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kUjzi~e_#KfU*y94a$a#&&pg3<^Gx{C)ik7Ks9Rn7KgdLaCIA2c literal 0 HcmV?d00001 diff --git a/src/main.typ b/src/main.typ index 719babf..415111c 100644 --- a/src/main.typ +++ b/src/main.typ @@ -271,9 +271,9 @@ Un'altra osservazione sul comportamento dell'isotopia regolare rispetto alle mos // C'è un'osservazione importante da fare quando parliamo di isotopia regolare. Quest'ultima è definita solo quando si parla di diagrammi. Quando invece parliamo di isotopia ambiente ha senso parlare dei link sia come diagrammi che come embedding $bb(S)^1 -> bb(R)^3$. -// C'è un'osservazione importante da fare quando parliamo di isotopia regolare. Quest'ultima è definita solo quando si parla di diagrammi nel piano o su $bb(S)^2$. Nel caso dell'isotopia regolare su $bb(S)^2$ questa è equivalente all'isotopia di _link con framing_ #footnote[Nel caso di nodi come embedding da $bb(S)^1 -> bb(S)^3$, un *nodo con framing* è un embedding di $bb(D)^2 times bb(S)^1 -> bb(S)^3$. Il *framing* del nodo è l'immagine di $I times bb(S)^1 -> bb(S)^3$. Un nodo con framing può essere visto come una banda embedded dove il framing è il numero di twist con segno.]. In particolare l'isotopia regolare preserva il framing del nodo. +C'è un'osservazione importante da fare quando parliamo di isotopia regolare. Quest'ultima è definita solo quando si parla di diagrammi nel piano o su $bb(S)^2$. Nel caso dell'isotopia regolare su $bb(S)^2$ questa è equivalente all'isotopia di _link con framing_ #footnote[Nel caso di nodi come embedding da $bb(S)^1 -> bb(S)^3$, un *nodo con framing* è un embedding di $bb(D)^2 times bb(S)^1 -> bb(S)^3$. Il *framing* del nodo è l'immagine di $I times bb(S)^1 -> bb(S)^3$. Un nodo con framing può essere visto come una banda embedded dove il framing è il numero di twist con segno.]. In particolare l'isotopia regolare preserva il framing del nodo. -Un'altra osservazione importante che possiamo fare quando si parla di isotopia regolare è che quest'ultima è equivalente all'isotopia di _link con framing_ #footnote[Nel caso di nodi come embedding da $bb(S)^1 -> bb(S)^3$, un *nodo con framing* è un embedding di $bb(D)^2 times bb(S)^1 -> bb(S)^3$. Il *framing* del nodo è l'immagine di $I times bb(S)^1 -> bb(S)^3$. Un nodo con framing può essere visto come una banda embedded dove il framing è il numero di twist con segno.]. In particolare l'isotopia regolare preserva il framing del nodo. +// Un'altra osservazione importante che possiamo fare quando si parla di isotopia regolare è che quest'ultima è equivalente all'isotopia di _link con framing_ #footnote[Nel caso di nodi come embedding da $bb(S)^1 -> bb(S)^3$, un *nodo con framing* è un embedding di $bb(D)^2 times bb(S)^1 -> bb(S)^3$. Il *framing* del nodo è l'immagine di $I times bb(S)^1 -> bb(S)^3$. Un nodo con framing può essere visto come una banda embedded dove il framing è il numero di twist con segno.]. In particolare l'isotopia regolare preserva il framing del nodo. Questo sarà importante più avanti quando vedremo invarianti come il polinomio $F_K$ che risulteranno essere invarianti per isotopia ambiente. Nel caso del polinomio $F_K$, il calcolo va fatto fissando un particolare diagramma del link. È tutt'ora un problema aperto dare una definizione "intrinseca" di $F_K$ senza usare diagrammi. @@ -395,7 +395,7 @@ Vedremo come questi assiomi definiscono un unico invariante di isotopia regolare *Notazione.* Per rendere alcuni calcoli più leggibili, in questo capitolo utilizzeremo in modo intercambiabile le notazioni $L_K, L[K]$ per indicare il polinomio di Kauffman associato ad un certo diagramma. #definition[ - Sia $K$ un diagramma di un link non orientato, e sia $bb(Z)[a, a^(-1), z, z^(-1)]$ l'anello i polinomi di Laurent sugli interi nelle variabili $a$ e $z$. + Sia $K$ un diagramma di un link non orientato, e sia $bb(Z)[a, a^(-1), z, z^(-1)]$ l'anello dei polinomi di Laurent sugli interi nelle variabili $a$ e $z$. Allora $L_K in bb(Z)[a, a^(-1), z, z^(-1)]$, e verifica i seguenti assiomi: @@ -417,7 +417,9 @@ Vedremo come questi assiomi definiscono un unico invariante di isotopia regolare // Queste sono le cosiddette relazioni skein del polinomio di Kauffman, come in precedenza ogni equazione sta a significare che ci sono dei diagrammi identici ovunque tranne nella parte evidenziata. -Come già anticipato in precedenza, vedremo che questo è un invariante di isotopia regolare e può essere generalizzato ad invariante di isotopia ambiente come segue +Al momento non sappiamo se il polinomio $L_K$ sia ben definito, ovvero se esiste un polinomio $L_K$ che verifica gli assiomi e non sappiamo neanche se sia unico. + +In questo lavoro vedremo che $L_K$ esiste ed è unico e che è un invariante di isotopia regolare e può essere generalizzato ad invariante di isotopia ambiente come segue #definition[ Definiamo $F_K in Z[a, a^(-1), z, z^(-1)]$ per diagrammi di link orientati $K$ come @@ -561,6 +563,10 @@ $ Un diagramma $K$ è una *somma connessa* se appare come due diagrammi disgiunti connessi da due archi paralleli (a meno di isotopie planari). Tagliando questi due archi e riunendo insieme le estremità dello stesso sotto-diagramma otteniamo un nuovo diagramma split della forma $tilde(K) = K_1 union.sq K_2$, in questo caso chiamiamo la somma connessa $K = K_1 hash K_2$. ] +#todo[aggiungere figura] + +In particolare abbiamo le seguenti identità per i polinomi $L_K$ e $F_K$ in relazione ai diagrammi _split_ e in _somma connessa_. La dimostrazione è omessa ma si fa procedendo per induzione sul numero di incroci. + #fact[ Valgono le seguenti relazioni di $L$ ed $F$ per diagrammi split o in somma connessa @@ -581,7 +587,7 @@ $ dove $delta = (a + a^(-1)) slash z - 1$ ed è lo stesso coefficiente che abbiamo trovato nell'esempio precedente. ] -== Strumenti Forma Induttiva +== Considerazioni preliminari Prima di descrivere la definizione induttiva diamo alcune definizioni per certe manipolazioni dei diagrammi di nodi e link. @@ -589,9 +595,9 @@ Prima di descrivere la definizione induttiva diamo alcune definizioni per certe Sia $K$ un diagramma di un nodo, $cal(U)$ la sua ombra planare#footnote[Ovvero per $K subset bb(R)^3$ poniamo $cal(U) colon.eq pi(K) subset bb(R)^2$] e $p in cal(U)$ un punto di partenza direzionato in $cal(U)$. ] -- Il suo *nodo banale standard* (o _in forma discendente_) associato a $K$ detto $hat(K)(cal(U), p)$ è definito come segue: si prende il primo punto di partenza direzionato e si inizia a percorrere l'ombra planare in quella direzione e si rende ogni incrocio un sopra-incrocio quando ci si passa sopra per la prima volta. +- Il suo *nodo banale standard* (o _in forma discendente_) associato a $K$ detto $hat(K)(cal(U), p)$ è definito come segue: si prende il punto di partenza direzionato e si inizia a percorrere l'ombra planare in quella direzione e si rende ogni incrocio un sopra-incrocio quando ci si passa sopra per la prima volta. - Questa definizione si estende anche a diagrammi di link prendendo un insieme di punti base ordinati come mostrato in figura. + Questa definizione si estende anche a diagrammi di link prendendo un insieme di punti base ordinati come mostrato in figura. In particolare questo induce un ordinamento dall'alto verso il basso delle componenti. #figure( image("assets/standard-unlink-construction.png", width: 12cm), @@ -600,7 +606,7 @@ Prima di descrivere la definizione induttiva diamo alcune definizioni per certe ], ) -- Definiamo ora le seguenti mosse su un diagramma, sia $i$ l'etichetta di un incrocio del diagramma +- Definiamo ora le seguenti mosse su un diagramma $K$, diamo un'etichetta ad ogni incrocio e sia $i$ l'etichetta di un incrocio del diagramma #{ set align(center) @@ -635,9 +641,9 @@ Prima di descrivere la definizione induttiva diamo alcune definizioni per certe in questo caso $S_i K$ scambia l'informazione sopra/sotto dell'incrocio, mentre $E_i K$ e $e_i K$ applicano gli *splice* orizzontali e verticali a quell'incrocio. -Sia $lambda = (lambda_n, dots, lambda_0)$ una sequenza di etichette di suoi incroci. +Sia $lambda = (lambda_n, dots, lambda_0)$ una sequenza di etichette di incroci di $K$. -- Definiamo le seguenti operazioni compatte per una sequenza di scambi seguita da uno splice finale +- Definiamo le seguenti notazioni compatte per una sequenza di scambi seguita da uno splice finale $ A_i^lambda colon.eq E_lambda_i S_lambda_(i-1) dots.c space S_lambda_0 @@ -645,7 +651,7 @@ Sia $lambda = (lambda_n, dots, lambda_0)$ una sequenza di etichette di suoi incr B_i^lambda colon.eq e_lambda_i S_lambda_(i-1) dots.c space S_lambda_0 $ - notiamo che per $i=0$ abbiamo che $A_0^lambda K = E_lambda_0 K$ e $B_0^lambda K = e_lambda_0 K$ che è l'unico caso degenere per questa definizione. + notiamo che per $i=0$ abbiamo $A_0^lambda K = E_lambda_0 K$ e $B_0^lambda K = e_lambda_0 K$ che è l'unico caso degenere per questa definizione. - Poniamo $hat(K)(lambda) = S_lambda_n dots.c space S_lambda_0 K$ e definiamo le notazioni compatte per le seguenti somme alternate @@ -655,37 +661,39 @@ Sia $lambda = (lambda_n, dots, lambda_0)$ una sequenza di etichette di suoi incr Omega_K (lambda) & colon.eq (-1)^(abs(lambda) + 1) L[hat(K)(lambda)] + z sum_K (lambda) $ - dove $abs(lambda) colon.eq n$ per $lambda$. + dove $abs(lambda) colon.eq n$. + +#fact[ + Un nodo banale standard è equivalente ad un nodo banale a meno di isotopia ambiente. +] *Osservazione.* Se consideriamo un diagramma di link $K$, $hat(K)(cal(U), p)$ ci induce una sequenza di scambi $lambda$ su $K$ tale che $hat(K)(cal(U), p) = S_lambda_n dots.c space S_lambda_0 K$. -*Osservazione.* Se abbiamo una sequenza di operazioni $S_i$, $E_i$, $e_i$ su incroci diversi queste operazioni commutano tra loro. Questo segue dal fatto che queste operazioni fanno solo modifiche locali al diagramma. +#observation[ + Se abbiamo una sequenza di operazioni $S_i$, $E_i$, $e_i$ su incroci diversi queste operazioni commutano tra loro. Questo segue dal fatto che queste operazioni fanno solo modifiche locali al diagramma. +] #definition[ - Dati $K_1, K_2$ diagrammi di link, diciamo che $K_1$ *sovrasta* $K_2$ se ogni incrocio tra $K_1$ e $K_2$ abbiamo che sono tutti sopra-incroci. + Dati $K_1, K_2$ diagrammi di link, diciamo che $K_1$ *sovrasta* $K_2$ se ogni incrocio in $K_1 inter K_2$ è un sopra-incrocio. ] -== Forma Induttiva +#align(center, line(stroke: 0.5pt, length: 50%)) -Per prima cosa enunciamo un lemma che utilizzeremo più volte in seguito. +Enunciamo ora un lemma che utilizzeremo più volte in seguito. #lemma[ Sia $K$ un diagramma di un nodo in forma discendente, allora o è composto interamente da riccioli o ammette una sequenza di mosse di tipo II e III che portano al diagramma di un nodo formato solo da riccioli. ] #proof[ - Per prima cosa osserviamo che $K$ come nodo è equivalente ad un nodo banale . Esisterà quindi una sequenza di mosse di tipi I, II, III che porta il diagramma $K$ al diagramma $#skein.unit$. Rimuovendo una alla volta da questa successione tutte le mosse di tipo I e modificando man mano le mosse II e III della successione in modo da far passare i riccioli attraverso i fili (come visto in precedenza si può fare usando alcune mosse II e III) otteniamo una successione composta solamente da mosse II e III che ci porta ad un diagramma composto solamente da riccioli. + #todo[da rifare, bla bla diagramma $D_1 -> dots -> D_n$ bla bla mosse $R_i$ di tipo I, II, III ... $D_1' -> dots -> D_n'$] ] -Vediamo ora l'idea che ci permette di trovare una formula chiusa per il polinomio $L_K$ che utilizzeremo nella dimostrazione della buona definizione. - -=== Caso base: nodo banale standard +Vediamo ora l'idea che ci permette di trovare una formula chiusa per il polinomio $L_K$ che utilizzeremo nella dimostrazione della buona definizione. Per prima cosa analizziamo quello che sarà il caso base della nostra definizione induttiva, ovvero quando abbiamo un nodo banale standard. Utilizzando il lemma precedente, possiamo calcolare esplicitamente $L_(hat(K)(cal(U), p))$ usando solo gli assiomi #link()[ii.b), ii.c), ii.d)] ovvero quelli per $#skein.over-twist-medium$, $#skein.under-twist-medium$ e $#skein.unit-medium$. Segue facilmente per induzione che $L_hat(K) (a, z) = a^w(K)$. -=== Caso ricorsivo - -Dato un diagramma di un nodo o link $K$, l'idea per la costruzione della definizione ricorsiva è di considerare il suo nodo o link discendente standard $hat(K)(cal(U),p )$ rispetto ad un punto base $p$. +Dato un diagramma di un nodo $K$, l'idea per la costruzione della definizione ricorsiva è di considerare il suo nodo o link discendente standard $hat(K)(cal(U), p)$ rispetto ad un punto base $p$. Questo induce una sequenza di indici $lambda$ che indichiamo con $lambda = (0, dots, n)$. Se applichiamo incrementalmente gli scambi a $K$, ovvero consideriamo $S_i dotss S_0 K$ al variare di $i$, ed applichiamo la relazione skein #link()[ii.a)] a questi diagrammi, otteniamo le seguenti relazioni: @@ -767,7 +775,7 @@ Da cui otteniamo un'espressione ricorsiva per $L_K$: Questo ci dà un'idea su come sia possibile calcolare induttivamente $L_K$ in termini di $L_hat(K)$ e di altri diagrammi con meno incroci del diagramma di partenza o più "vicini" a $hat(K)$. -Affinché sia sempre possibile raggiungere un nodo in forma di nodo banale standard serve anche aggiungere la relazione +Affinché l'induzione possa sempre raggiungere il caso base di un nodo in forma discendente serve anche aggiungere la relazione $ L_(K_1 union.sq K_2) = delta L_(K_1) L_(K_2) @@ -779,7 +787,7 @@ sempre con $delta = (a + a^(-1)) slash z - 1$. === La definizione induttiva -#definition(name: [Forma chiusa per $L_K$])[ +#definition(name: [Formula chiusa per $L_K$])[ Il polinomio $kL_(K)(a,z)$ è definito induttivamente come segue, abbiamo i seguenti casi: ] @@ -791,22 +799,22 @@ sempre con $delta = (a + a^(-1)) slash z - 1$. kL_K (a, z) colon.eq a^w(K) $ -2. Se $K = K_1 union K_2$ con $K_1$ diagramma di un nodo e $K_2$ diagramma di link ed abbiamo che $K_1$ _sovrasta_ $K_2$, allora poniamo +2. Se $K = K_1 union K_2$ con $K_1$ diagramma di un nodo e $K_2$ diagramma di link e $K_1$ _sovrasta_ $K_2$, allora poniamo $ - kL(K_1 union K_2) colon.eq d kL(K_1) kL(K_2) + kL(K_1 union K_2) colon.eq delta kL(K_1) kL(K_2) $ - con $d colon.eq (a + a^(-1)) slash z - 1$. + con $delta colon.eq (a + a^(-1)) slash z - 1$. -3. Sia $K = K_1 union dotss union K_n$ e nessun $K_i$ sovrasta tutti gli altri#footnote[Questo sicuramente accade altrimenti potremmo usare il punto ii).] allora: +3. Sia $K = K_1 union dotss union K_n$ con $K_i$ diagrammi di nodi e tale che nessun $K_i$ sovrasta tutti gli altri allora: #[ #set enum(numbering: "a)") 1. #marker([iii.a)]) - Se $K$ è composto da più componenti allora siano $p_1, ..., p_n$ dei punti di partenza direzionati su $K_1, ..., K_n$ e $overline(p)_i$ lo stesso punto di partenza $p_i$ con la direzione opposta su $K_i$. Sia $lambda(p_i)$ la sequenza di scambi di incroci di $K_i$ con $K - K_i$ tale che $hat(K)(lambda(p_i)) = K_i union.sq (K - K_i)$ tale che $K_i$ sia _sovrastante_ al resto delle componenti (chiameremo questa _sequenza di rialzo_). A questo punto possiamo definire $kL_K$ come + Se $n > 1$ allora siano $p_1, ..., p_n$ dei punti di partenza direzionati su $K_1, ..., K_n$ e $overline(p)_i$ lo stesso punto di partenza $p_i$ con la direzione opposta su $K_i$. Sia $lambda(p_i)$ la sequenza di scambi di incroci tra $K_i$ e $K - K_i$ tale che $hat(K)(lambda(p_i)) = K_i union.sq (K - K_i)$, con $K_i$ _sovrastante_ al resto delle componenti (chiameremo questa _sequenza di rialzo_). A questo punto possiamo definire $L_K$ come #[ #set text(size: 11pt) @@ -814,12 +822,12 @@ sempre con $delta = (a + a^(-1)) slash z - 1$. L_K (a, z) colon.eq 1 / (2n) [ sum_(i=1)^n sum_(q=p_i, overline(p)_i) - ((-1)^(abs(lambda(q))+1) d kL_(K_i) kL_(K - K_i) + z sum_K (lambda(q))) + ((-1)^(abs(lambda(q))+1) delta L_(K_i) L_(K - K_i) + z sum_K (lambda(q))) ] $ ] - 2. #marker([iii.b)]) Se $K$ è una singola componente allora sia $p$ un punto di partenza direzionato su $K$ e $overline(p)$ quello con direzione opposta. Sia $lambda(p)$ la sequenza di scambi di incroci che lo porta al nodo banale standard $hat(K)$ e definiamo + 2. #marker([iii.b)]) Se $n=1$ allora sia $p$ un punto di partenza direzionato su $K$ e $overline(p)$ quello con direzione opposta. Sia $lambda(p)$ la sequenza di scambi di incroci che lo porta al nodo banale standard $hat(K)$ e definiamo #[ #set text(size: 11pt) @@ -828,7 +836,7 @@ sempre con $delta = (a + a^(-1)) slash z - 1$. L_K (a, z) colon.eq 1 / 2 [ sum_(q = p, overline(p)) - ((-1)^(abs(lambda(q))+1) kL(hat(K)(lambda(q))) + z sum_K (lambda(q))) + ((-1)^(abs(lambda(q))+1) L_(hat(K)(q)) + z sum_K (lambda(q))) ] $ ] @@ -845,11 +853,11 @@ Nel corso della dimostrazione tutti gli argomenti per induzione si baseranno sul #definition( name: [L'ipotesi induttiva], )[ - L'ipotesi induttiva che useremo nel corso della dimostrazione è la la seguente: + L'ipotesi induttiva che useremo nel corso della dimostrazione è la seguente: #set enum(numbering: "a)") - Per ogni diagramma di link $K$ con $< N$ incroci e diagrammi $#skein.over-twist-medium$ con meno di $N$ incroci: + Per ogni diagramma di link $K$ con $< N$ incroci e per diagrammi contenenti $#skein.over-twist-medium$ con meno di $N$ incroci: 1. $L_K$ è ben definito (non dipende dalla scelta di punto base). @@ -859,7 +867,7 @@ Nel corso della dimostrazione tutti gli argomenti per induzione si baseranno sul - $L[#skein.over-twist-medium] = a L [#skein.strand-medium]$ - - $L[#skein.under-twist-medium] = a L [#skein.strand-medium]$ + - $L[#skein.under-twist-medium] = a^(-1) L [#skein.strand-medium]$ 3. $L_K$ è invariante per mosse di tipo II e III che non aumentano il numero di incroci. ] @@ -924,7 +932,10 @@ Nel corso della dimostrazione tutti gli argomenti per induzione si baseranno sul + (-1)^(n+1) ( & L[E_0 S_n dotss S_1 K] + L[e_0 S_n dotss S_1 K] ) $ - Permutando i termini di tutte le righe centrali possiamo riscriverli in modo da averli nella forma $L[#blank K] + L[#blank S_0 K]$: + Permutando i termini di tutte le righe centrali possiamo + // riscriverli in modo da averli nella forma + ottenere una somma di termini della forma + $L[blank K] + L[blank S_0 K]$: $ sum_K (lambda) - sum_K (mu) = @@ -936,10 +947,22 @@ Nel corso della dimostrazione tutti gli argomenti per induzione si baseranno sul + (-1)^(n+1) ( & L[E_0 S_n dotss S_1 K] + L[e_0 S_n dotss S_1 K] ) $ - Ora utilizziamo un'osservazione precedente riguardo l'ordine di applicazione delle operazioni da cui segue che $S_(n-1) dots.c space.med S_1 S_0 K = S_0 S_(n-1) dots.c space.med S_1 K$, dunque possiamo applicare l'induzione sostituendo: + Ora ricordiamo l'@observation-seq-ops-commute per cui operazioni su indici diversi commutano tra loro, quindi possiamo spostare l'operazione $S_0$ in modo che sia applicata per ultima ed otteniamo: + + $ + sum_K (lambda) - sum_K (mu) = + + ( & L[E_0 K] + L[e_0 K]) + \ + - ( & L[E_1 K] + L[S_0 E_1 K] + L[e_1 K] + L[S_0 e_1 K] ) + \ + space dots.v \ + (-1)^n ( & L[E_n S_(n-1) dotss S_1 K ] + L[S_0 E_n S_(n-1) dotss S_1 K ] + \ + & + L[e_n S_(n-1) dotss S_1 K ] + L[S_0 e_n S_(n-1) dotss S_1 K ] ) \ + + (-1)^(n+1) ( & L[E_0 S_n dotss S_1 K] + L[e_0 S_n dotss S_1 K] ) + $ + + A questo punto notiamo che tutti i diagrammi $E_i S_(i-1) dotss S_1 K$ e $e_i S_(i-1) dotss S_1 K$ per $i = 1, dots, n$ hanno applicato uno splice quindi hanno strettamente meno incroci di $K$. Possiamo quindi applicare l'ipotesi induttiva e sostituire tutti i termini nella forma: $ - L[#blank] + L[S_i #blank] = z (L [E_i #blank] + L[e_i #blank]) + L[blank] + L[S_0 blank] = z (L [E_0 blank] + L[e_0 blank]) $ $ @@ -952,7 +975,7 @@ Nel corso della dimostrazione tutti gli argomenti per induzione si baseranno sul + (-1)^(n+1) ( & L[E_0 S_n dotss S_1 K] + L[e_0 S_n dotss S_1 K] ) $ - Riordiniamo ora le operazioni e portiamo $E_0$ e $e_0$ all'inizio delle sequenze di operazioni: + Riordiniamo ora le operazioni in modo da avere $E_0$ e $e_0$ applicate per prime nelle sequenze di operazioni: #let hl(c: color.mix((white, 100%), (black, 10%)), content) = box(radius: 2pt, outset: (y: 3pt, x: 1pt), fill: c, content) @@ -965,9 +988,7 @@ Nel corso della dimostrazione tutti gli argomenti per induzione si baseranno sul + (-1)^(n+1) ( & L[ S_n dotss S_1 #hl($E_0 K$)] + L[ S_n dotss S_1 #hl($e_0 K$)] ) $ - #colbreak() - - Ora distribuiamo tutti i termini che riguardano $E_0 K$ e poi tutti quelli per $e_0 K$ ed otteniamo quanto segue: + Ora distribuiamo e riorganizziamo tutti i termini in modo da avere prima quelli che riguardano $E_0 K$ e poi tutti quelli per $e_0 K$ ed otteniamo quanto segue: $ = L[ #hl($E_0 K$)] &+ (-1)^(n+1) L[S_n dotss S_1 #hl($E_0 K$)] + \ @@ -982,14 +1003,14 @@ Nel corso della dimostrazione tutti gli argomenti per induzione si baseranno sul +( -1)^n ( & L[E_n S_(n-1) dots S_1 #hl($e_0 K$)] + L[e_n S_(n-1) dots S_1 #hl($e_0 K$)]))) $ - Infine osserviamo che i termini in $z(#blank)$ sono proprio le somme incrementali sugli scambi per i diagrammi $E_0 K$ e $e_0 K$ per le sequenze $(n, dots, 1)$, dunque possiamo riscrivere l'espressione precedente come + Infine osserviamo che i termini dentro le parentesi in $z(#blank)$ sono proprio somme della forma $sum_(E_0 K) (gamma)$ e $sum_(e_0 K) (gamma)$ per $gamma = (n, dots, 1)$, dunque possiamo riscrivere l'espressione precedente come $ - = L[ E_0 K] &+ (-1)^(n+1) L[S_n dotss S_1 E_0 K] - z sum_(E_0 K) ((n, dots, 1)) \ - + L[ e_0 K] &+ (-1)^(n+1) L[S_n dotss S_1 e_0 K] - z sum_(e_0 K) ((n, dots, 1)) + = L[ E_0 K] &+ (-1)^(n+1) L[S_n dotss S_1 E_0 K] - z sum_(E_0 K) (gamma) \ + + L[ e_0 K] &+ (-1)^(n+1) L[S_n dotss S_1 e_0 K] - z sum_(e_0 K) (gamma) $ - Ora possiamo applicare l'ipotesi induttiva in quanto tutti i diagrammi hanno strettamente meno di $N$ incroci. Applicando l'ipotesi induttiva per $gamma = (n, dots, 1)$ e osservando che con la convenzione di Kauffman $abs(gamma) + 1 = (n - 1) + 1 = n$, otteniamo che + Ora possiamo applicare l'ipotesi induttiva in quanto tutti i diagrammi hanno strettamente meno di $N$ incroci. Applicando l'ipotesi induttiva per $gamma$ e osservando che con la convenzione di Kauffman $abs(gamma) + 1 = (n - 1) + 1 = n$, otteniamo che $ Omega_K(gamma) @@ -1001,8 +1022,7 @@ Nel corso della dimostrazione tutti gli argomenti per induzione si baseranno sul infine sostituendo $ - &= L[E_0 K] - Omega_(E_0 K) (gamma) + L[e_0 K] - Omega_(e_0 K) (gamma) \ - &= 0 + 0 = 0 + = L[E_0 K] - Omega_(E_0 K) (gamma) + L[e_0 K] - Omega_(e_0 K) (gamma) = 0 + 0 = 0 $ E questo conclude la dimostrazione. @@ -1017,295 +1037,370 @@ Nel corso della dimostrazione tutti gli argomenti per induzione si baseranno sul L_K (a, z) colon.eq 1 / (2n) [ sum_(i=1)^n sum_(q=p_i, overline(p)_i) - ((-1)^(abs(lambda(q))+1) d kL_(K_i) kL_(K - K_i) + z sum_K (lambda(q))) + ((-1)^(abs(lambda(q))+1) delta kL_(K_i) kL_(K - K_i) + z sum_K (lambda(q))) ] $ - le espressioni che dipendono dai punti base sono $(-1)^(abs(lambda(q)) + 1)$ e le somme sugli scambi $sum_K (lambda(q))$. Se applichiamo una $1$-rotazione alle sequenze di rialzo $lambda(q)$ le lunghezze non cambiano ed abbiamo appena mostrato che $sum_K (lambda(q)))$ è invariante per $1$-rotazioni. + le espressioni che dipendono dai punti base sono $(-1)^(abs(lambda(q)) + 1)$ e $sum_K (lambda(q))$. Se applichiamo una $1$-rotazione alle sequenze di rialzo $lambda(q)$ le lunghezze non cambiano ed usando il @lemma-sum-switches-rotation anche le somme $sum_K (lambda(q)))$ sono invarianti per $1$-rotazioni. A questo punto possiamo estendere l'invarianza per $1$-rotazioni a tutte le possibili permutazioni cicliche e questo ci dimostra che questo caso non dipende dalla scelta del punto base. ] #align(center, line(stroke: 0.5pt, length: 50%)) -// Ora rimane da verificare l'invarianza per scelta di punto base per il caso ii.b) ovvero quando $K$ è fo per $p$ punto base direzionato su $K$. +Ora rimane da verificare l'invarianza per scelta di punto base per il caso @kauffman-rec-single-component ovvero quando $K$ è formato da una sola componente, ricordiamo l'espressione: -// $ -// L_K (a, z) colon.eq -// 1 / 2 [ -// sum_(q = p, overline(p)) -// ((-1)^(abs(lambda(q))+1) L[hat(K)(lambda(q))] + z sum_K (lambda(q))) -// ] -// $ +$ + L_K (a, z) colon.eq + 1 / 2 [ + sum_(q = p, overline(p)) + ((-1)^(abs(lambda(q))+1) L[hat(K)(lambda(q))] + z sum_K (lambda(q))) + ] +$ -// #lemma[ -// Consideriamo i due modi di fare uno splice di un nodo banale standard al primo incrocio subito dopo un punto base direzionato. Ci sono due casi: in uno otteniamo due nodi banali in forma standard, nell'altro un solo nodo banale (non necessariamente in forma standard). -// ] +La dimostrazione procede sempre mostrando l'indipendenza dalla scelta di punto base. Vediamo alcuni fatti: -// #proof[ -// #todo[work in progress] -// ] +#grid( + columns: (1fr, auto), + gutter: 1em, + fact[ + Consideriamo i due modi di fare uno splice di un nodo banale standard al primo incrocio subito dopo un punto base direzionato. Ci sono due casi: in uno otteniamo _due nodi banali in forma standard_, nell'altro _un solo nodo banale_ (non necessariamente in forma discendente). + ], + image("assets/splice-circuits.png", width: 4cm), +) // #lemma[ -// La proprietà del lemma @std-unknot-to-curls si estende al diagramma $e_i hat(K)$. -// ] - -// #proof[ -// #todo[work in progress] +// Consideriamo i due modi di fare uno splice di un nodo banale standard al primo incrocio subito dopo un punto base direzionato. Ci sono due casi: in uno otteniamo due nodi banali in forma standard, nell'altro un solo nodo banale (non necessariamente in forma standard). // ] -// Vediamo ora alcune identità per nodi banali standard che ci mostrano cosa succede quando scorriamo il punto base di un incrocio nella direzione del punto base. +// _Idea Dimostrazione._ Nel caso in cui si formano due componenti, se studiamo come sono fatti gli incroci di ogni componente con se stessa notiamo che rispettano ancora la proprietà della forma discendente e poiché tutti gli incroci della seconda componente con la "prima parte" della prima componente sono sotto-incroci e con la "seconda parte" sono sopra-incroci possiamo applicare mosse II e III per separare le due componenti. -// #lemma[ -// Sia $p$ un punto di partenza direzionato e $hat(K) colon.eq hat(K)(cal(U), p)$ un nodo banale standard. Sia $i$ il primo incrocio in $K$ subito dopo il punto di partenza $p$. Sia $q$ un altro punto base direzionato posto nell'arco subito dopo l'incrocio $i$ con la sua stessa direzione e poniamo $hat(K)' colon.eq hat(K)(cal(U), q)$. Allora valgono le seguenti proprietà - -// 1. $S_i hat(K) = hat(K)'$ +// Nel caso di una sola componente non abbiamo più un nodo in forma discendente, ma questa è formata da una prima parte discendente, seguita da una ascendente (la parte in mezzo è percorsa al contrario) ed infine un'altra discendente, pensando in termini di altezze rispetto al piano di proiezione, ogni parte può essere semplificata indipendentemente dalle altre quindi abbiamo ancora un nodo banale a meno di isotopia ambiente. -// 2. $L[hat(K)] + L[hat(K)'] = z (L[E_i hat(K)] + L[e_i hat(K)])$ -// ] - -// #proof[ -// 1. Per definizione di nodo banale standard $hat(K)(cal(U), q) = S_i hat(K)(cal(U), p)$ poiché l'incrocio $i$ in $hat(K)(cal(U), p)$ compare come sopra-incrocio mentre in $hat(K)(cal(U), q)$ è l'ultimo incrocio visitato quindi è certamente la seconda occorrenza di $i$ quindi è un sotto-incrocio. Tutti gli altri incroci compaiono nello stesso ordine quindi restano invariati. +#fact[ + La proprietà del lemma @std-unknot-to-curls si estende al diagramma $e_i hat(K)$. +] -// 2. Per questo secondo punto usiamo che $L_hat(K) = a^w(hat(K))$, assumiamo ora che tra $E_i hat(K)$ e $e_i hat(K)$ il primo sia il caso con due componenti composte da nodi banali standard ed il secondo quello con una sola componente di un nodo banale. +La dimostrazione di questi fatti è omessa in quanto molto tecnica, ma per dare un'idea di ciò che avviene in questi tre casi possiamo considerare la seguente figura. -// Supponiamo che $w(e_i hat(K)) = w$, allora in quanto abbiamo solo rimosso un incrocio necessariamente +#figure( + image("assets/std-unknot-splices.png", width: 12cm), + caption: [Diagramma di $K$, $E_i K$ e $e_i K$ con rispettivo grafico delle "altezze" dei fili.], +) -// $ -// {w(hat(K)), w(S_i hat(K))} = {w-1, w+1} -// $ +Nel primo caso il diagramma $K$ nodo è in forma discendente, possiamo visualizzarlo al variare "dell'altezza" come sotto. Dopo gli splice in un caso otteniamo due componenti disgiunte ognuna ancora in forma discendente, mentre nell'altro caso anche se abbiamo una parte rovesciata complessivamente il nodo rimane banale. -// ovvero i writhe di $hat(K)$ e $S_i hat(K)$ differiscono di $2$, assumiamo di essere nel caso $w(hat(K)) = w - 1$. +// #proof[ +// #todo[work in progress] +// ] -// Analogamente $E_i hat(K)$ avrà due componenti dunque $K = K_1 hash K_2$, ognuna con writhe rispettivamente $w_1$ e $w_2$. Per il teorema della curva di Jordan, la somma dei segni degli incroci tra componenti diverse ha somma zero poiché una delle due _sovrasta_ l'altra dunque $w_1 + w_2 = w$. +Vediamo ora alcune identità per nodi banali standard che ci mostrano cosa succede quando facciamo avanzare il punto base di un incrocio. -// $ -// L[hat(K)] &= a^(w+1) \ -// L[S_i hat(K)] &= a^(w-1) \ -// L[e_i hat(K)] &= a^w \ -// L[E_i hat(K)] &= d a^(w_1) a^(w_2) = d a^(w_1 + w_2) = d a^w = (a^(w+1) + a^(w-1)) slash z - a^w -// $ +#lemma[ + Sia $p$ un punto di partenza direzionato e $hat(K) colon.eq hat(K)(cal(U), p)$ un nodo banale standard. Sia $i$ il primo incrocio in $K$ subito dopo il punto di partenza $p$. Sia $q$ un altro punto base direzionato posto nell'arco subito dopo l'incrocio $i$ con la sua stessa direzione e poniamo $hat(K)' colon.eq hat(K)(cal(U), q)$. Allora valgono le seguenti proprietà -// che dunque verifica l'identità: + 1. $S_i hat(K) = hat(K)'$ -// $ -// a^(w+1) + a^(w-1) &= cancel(z) ((a^(w+1) + a^(w-1))/cancel(z) - cancel(a^w) + cancel(a^w)) \ -// => L[hat(K)] + L[hat(K)'] &= z (L[E_i hat(K)] + L[e_i hat(K)]) \ -// $ + 2. $L[hat(K)] + L[hat(K)'] = z (L[E_i hat(K)] + L[e_i hat(K)])$ +] -// E questo conclude la dimostrazione delle due identità. -// ] +#proof[ + 1. Per definizione di nodo banale standard $hat(K)(cal(U), q) = S_i hat(K)(cal(U), p)$ poiché l'incrocio $i$ in $hat(K)(cal(U), p)$ compare come sopra-incrocio mentre in $hat(K)(cal(U), q)$ è l'ultimo incrocio visitato quindi è certamente la seconda occorrenza di $i$ quindi è un sotto-incrocio. Tutti gli altri incroci compaiono nello stesso ordine quindi restano invariati. -// Possiamo ora mostrare l'invarianza per scelta di punto base di $Omega_K (p)$. + 2. Per questo secondo punto usiamo che $L_hat(K) = a^w(hat(K))$, assumiamo ora che tra $E_i hat(K)$ sia il caso con due componenti di due nodi banali standard e $e_i hat(K)$ quello con una sola componente di diagramma di nodo banale. -// #lemma[ -// Sia $K$ un diagramma di un nodo. Allora @kauffman-rec-single-component non dipende dalla scelta di punto base. Inoltre se $p$ è un punto base direzionato su $K$ e $lambda(p)$ la sequenza di scambi determinata da $p$ allora + Supponiamo che $w(e_i hat(K)) = w$, allora in quanto abbiamo solo rimosso un incrocio necessariamente -// $ -// Omega_K (p) = (-1)^(abs(lambda(p)) + 1) L_(hat(K)(p)) + z sum_K (lambda(p)) -// $ + $ + {w(hat(K)), w(S_i hat(K))} = {w-1, w+1} + $ -// non dipende dalla scelta di punto base. -// ] + ovvero i writhe di $hat(K)$ e $S_i hat(K)$ differiscono di $2$, assumiamo di essere nel caso $w(hat(K)) = w + 1$. -// #proof[ -// Possiamo assumere che la sequenza di scambi determinata da $p$ sia etichetta $lambda = (n, dots, 0)$, in questo modo + Analogamente $E_i hat(K)$ avrà due componenti dunque $K = K_1 hash K_2$, ognuna con writhe rispettivamente $w_1$ e $w_2$. Per il teorema della curva di Jordan, la somma dei segni degli incroci tra componenti diverse ha somma zero poiché una delle due _sovrasta_ l'altra dunque $w_1 + w_2 = w$. -// $ -// Omega_K (p) = (-1)^(n+1) L_(hat(K)(p)) + z sum_K (lambda(p)) -// $ + $ + L[hat(K)] &= a^(w+1) \ + L[S_i hat(K)] &= a^(w-1) \ + L[e_i hat(K)] &= a^w \ + L[E_i hat(K)] &= delta a^(w_1) a^(w_2) = delta a^(w_1 + w_2) = delta a^w = (a^(w+1) + a^(w-1)) slash z - a^w + $ -// Ricordiamo come era definito $L_K$ nel caso di una sola componente @kauffman-rec-single-component e notiamo quanto segue + che dunque verifica l'identità: -// $ -// L_K (a, z) &colon.eq 1 / 2 [ -// sum_(q = p, overline(p)) -// ((-1)^(abs(lambda(q))+1) kL(hat(K)(lambda(q))) + z sum_K (lambda(q))) -// ] \ -// &= 1/2 (Omega_K (p) + Omega_K (overline(p))) -// $ + $ + a^(w+1) + a^(w-1) &= cancel(z) ((a^(w+1) + a^(w-1))/cancel(z) - cancel(a^w) + cancel(a^w)) \ + => L[hat(K)] + L[hat(K)'] &= z (L[E_i hat(K)] + L[e_i hat(K)]) \ + $ -// Dunque ci basta mostrare che $Omega_K (p)$ non dipenda dalla scelta di punto base. + E questo conclude la dimostrazione delle due proprietà. +] -// Come in precedenza possiamo mostrarlo facendo scorrere $p$ nell'arco successivo al primo incrocio dopo $p$ (questo ci induce l'invarianza per permutazioni cicliche e ci permette di concludere). +Possiamo ora mostrare l'invarianza per scelta di punto base di $Omega_K (p)$. -// Sia $i$ l'etichetta del primo incrocio dopo $p$. In $hat(K)(p)$ questo incrocio sarà sicuramente un sopra-incrocio in quanto $i$ è visitato per la prima volta essendo il primo incrocio dopo $p$. Invece in $K(p)$ può essere sia un sopra-incrocio che un sotto-incrocio, abbiamo quindi due casi: +#lemma[ + Sia $K$ un diagramma di un nodo. Allora @kauffman-rec-single-component non dipende dalla scelta di punto base. Inoltre se $p$ è un punto base direzionato su $K$ e $lambda(p)$ la sequenza di scambi determinata da $p$ che lo porta a $hat(K)(p) colon.eq hat(K)(cal(U), p)$ allora -// - Se $i$ è un sotto-incrocio in $K(p)$ allora $i$ sarà il primo incrocio nella sequenza di scambi ed avremo $i = n$ + $ + Omega_K (p) = (-1)^(abs(lambda(p)) + 1) L_(hat(K)(p)) + z sum_K (lambda(p)) + $ -// #todo[disegnino] + non dipende dalla scelta di punto base. +] -// Consideriamo ora la situazione di $K(q)$ +#proof[ + Possiamo assumere che la sequenza di scambi determinata da $p$ sia etichetta $lambda = (n, dots, 0)$, in questo modo -// #todo[disegnino] + $ + Omega_K (p) = (-1)^(n+1) L_(hat(K)(p)) + z sum_K (lambda(p)) + $ -// come già detto prima se l'ultimo incrocio è un sotto-incrocio allora è già in forma di nodo banale standard. Segue che $i$ non appartiene alla sequenza di scambi in questo caso ed avremo $(n-1, dots, 0)$ e $hat(K)(q) = S_(n-1) dotss S_0 K$. Vorremo vedere che $Omega_K (p) = Omega_K (q)$, ovvero: + Ricordiamo come era definito $L_K$ nel caso di una sola componente @kauffman-rec-single-component e notiamo quanto segue -// $ -// Omega_K (p) &= (-1)^(n+1) L[hat(K)(p)] + z sum_K (lambda(p)) \ -// Omega_K (q) &= (-1)^((n-1)+1) L[hat(K)(q)] + z sum_K (lambda(q)) -// $ + $ + L_K (a, z) &colon.eq 1 / 2 [ + sum_(q = p, overline(p)) + ((-1)^(abs(lambda(q))+1) L_(hat(K)(q)) + z sum_K (lambda(q))) + ] \ + &= 1/2 (Omega_K (p) + Omega_K (overline(p))) + $ -// come in precedenza studiamo la differenza: + Dunque ci basta mostrare che $Omega_K (p)$ non dipenda dalla scelta di punto base. -// $ -// Omega_K (p) - Omega_K (q) -// =& (-1)^(n+1) L[hat(K)(p)] + z sum_K (lambda(p)) \ -// &+ underbrace((-1)^(n+1), = -(-1)^n) L[hat(K)(q)] - z sum_K (lambda(q)) \ -// =& (-1)^(n+1) (L[hat(K)(p)] + L[hat(K)(q)]) \ -// &+ z (-1)^n (L[A_n^lambda K] + L[B_n^lambda K]) -// $ + Come in precedenza possiamo mostrarlo facendo scorrere $p$ nell'arco successivo al primo incrocio dopo $p$ (questo ci induce l'invarianza per permutazioni cicliche e ci permette di concludere). -// dove abbiamo usato il fatto che tutti i termini in $sum_K (lambda(p)) - sum_K (lambda(q))$ si cancellano tra loro tranne l'ultimo di $sum_K (lambda(p))$. + Sia $i$ l'etichetta del primo incrocio dopo $p$. In $hat(K)(p)$ questo incrocio sarà sicuramente un sopra-incrocio in quanto $i$ è visitato per la prima volta essendo il primo incrocio dopo $p$. Invece in $K(p)$ può essere sia un sopra-incrocio che un sotto-incrocio, abbiamo quindi due casi: -// Ora notiamo che + #{ + set align(center) + grid( + columns: 2, + row-gutter: 0.5em, + column-gutter: 2em, + image("assets/derived/atlas-crossing-slides-0.png", width: 2.5cm), + image("assets/derived/atlas-crossing-slides-1.png", width: 2.5cm), + $K(p)$, + $K(p)$, + ) + } -// $ -// A_n^lambda K &= E_n S_(n-1) dotss S_0 K = E_n hat(K)(p) \ -// B_n^lambda K &= e_n S_(n-1) dotss S_0 K = e_n hat(K)(p) \ -// hat(K)(p) &= S_n space.med hat(K)(p) -// $ + - Se $i$ è un sotto-incrocio in $K(p)$ allora $i$ sarà il primo incrocio nella sequenza di scambi ed avremo $i = n$ + + #{ + set align(center) + grid( + columns: 2, + row-gutter: 0.5em, + column-gutter: 2em, + image("assets/derived/atlas-crossing-slides-1.png", width: 2.5cm), + image("assets/derived/atlas-crossing-slides-0.png", width: 2.5cm), + $K(p)$, + $hat(K)(p)$, + ) + } + + Consideriamo ora la situazione di $K(q)$: + + #{ + set align(center) + grid( + columns: 2, + row-gutter: 0.5em, + column-gutter: 2em, + image("assets/derived/atlas-crossing-slides-3.png", width: 2.5cm), + image("assets/derived/atlas-crossing-slides-3.png", width: 2.5cm), + $K(q)$, + $hat(K)(q)$, + ) + } + + come già detto prima se l'ultimo incrocio è un sotto-incrocio allora è già in forma di nodo banale standard. Segue che $i$ non appartiene alla sequenza di scambi in questo caso ed avremo $(n-1, dots, 0)$ e $hat(K)(q) = S_(n-1) dotss S_0 K$. Vorremo vedere che $Omega_K (p) = Omega_K (q)$, ovvero: -// inoltre per il @lemma-slide-identities abbiamo anche che + $ + Omega_K (p) &= (-1)^(n+1) L[hat(K)(p)] + z sum_K (lambda(p)) \ + Omega_K (q) &= (-1)^((n-1)+1) L[hat(K)(q)] + z sum_K (lambda(q)) + $ -// $ -// L[hat(K)(p)] + L[hat(K)(q)] = z (L[E_n hat(K)(p)] + L[e_n hat(K)(p)]) -// $ + come in precedenza studiamo la differenza: -// Per concludere questo caso basta sostituire in queste ultime identità + $ + Omega_K (p) - Omega_K (q) + =& (-1)^(n+1) L[hat(K)(p)] + z sum_K (lambda(p)) \ + &+ underbrace((-1)^(n+1), = -(-1)^n) L[hat(K)(q)] - z sum_K (lambda(q)) \ + =& (-1)^(n+1) (L[hat(K)(p)] + L[hat(K)(q)]) \ + &+ z (-1)^n (L[A_n^lambda K] + L[B_n^lambda K]) + $ -// $ -// Omega_K (p) - Omega_K (q) =& (-1)^(n+1) (L[hat(K)(p)] + L[hat(K)(q)]) \ -// &+ z (-1)^n (L[A_n^lambda K] + L[B_n^lambda K]) \ -// = & cancel(z (-1)^(n+1) (L[E_n hat(K)(p)] + L[e_n hat(K)(p)])) \ -// &+ cancel(z (-1)^n (L[E_n hat(K)(p)] + L[e_n hat(K)(p)])) \ -// = & 0 -// $ + dove abbiamo usato il fatto che tutti i termini in $sum_K (lambda(p)) - sum_K (lambda(q))$ si cancellano tra loro tranne l'ultimo di $sum_K (lambda(p))$. -// e quindi $Omega_K (p) = Omega_K (q)$. + Ora notiamo che -// - Se $i$ è un sopra-incrocio in $K(p)$ siamo nella seguente situazione + $ + A_n^lambda K &= E_n S_(n-1) dotss S_0 K = E_n hat(K)(p) \ + B_n^lambda K &= e_n S_(n-1) dotss S_0 K = e_n hat(K)(p) \ + hat(K)(p) &= S_n space.med hat(K)(p) + $ -// #todo[disegnino] + inoltre per il @lemma-slide-identities abbiamo anche che -// In questo caso, $i$ non fa parte della sequenza di scambi per $K(q)$. Però dopo aver spostato il punto base avremo la seguente situazione + $ + L[hat(K)(p)] + L[hat(K)(q)] = z (L[E_n hat(K)(p)] + L[e_n hat(K)(p)]) + $ -// #todo[disegnino] + Per concludere questo caso basta sostituire in queste ultime identità -// Questa volta $i$ fa parte della sequenza di scambi per $K(q)$ e quando compare come ultimo incrocio è un sotto-incrocio. Avremo quindi le seguenti sequenze di scambi + $ + Omega_K (p) - Omega_K (q) =& (-1)^(n+1) (L[hat(K)(p)] + L[hat(K)(q)]) \ + &+ z (-1)^n (L[A_n^lambda K] + L[B_n^lambda K]) \ + = & cancel(z (-1)^(n+1) (L[E_n hat(K)(p)] + L[e_n hat(K)(p)])) \ + &+ cancel(z (-1)^n (L[E_n hat(K)(p)] + L[e_n hat(K)(p)])) \ + = & 0 + $ -// $ -// lambda(q) &= (n, n-1, dots, i, dots, 1, 0) \ -// lambda(p) &= (n, n-1, dots, i+1, i-1, dots, 1, 0) -// $ + e quindi $Omega_K (p) = Omega_K (q)$. + + - Se $i$ è un sopra-incrocio in $K(p)$ siamo nella seguente situazione + + #{ + set align(center) + grid( + columns: 2, + row-gutter: 0.5em, + column-gutter: 2em, + image("assets/derived/atlas-crossing-slides-0.png", width: 2.5cm), + image("assets/derived/atlas-crossing-slides-0.png", width: 2.5cm), + $K(p)$, + $hat(K)(p)$, + ) + } + + In questo caso, $i$ non fa parte della sequenza di scambi per $K(q)$. Però dopo aver spostato il punto base avremo la seguente situazione + + #{ + set align(center) + grid( + columns: 2, + row-gutter: 0.5em, + column-gutter: 2em, + image("assets/derived/atlas-crossing-slides-2.png", width: 2.5cm), + image("assets/derived/atlas-crossing-slides-3.png", width: 2.5cm), + $K(q)$, + $hat(K)(q)$, + ) + } + + Questa volta $i$ fa parte della sequenza di scambi per $K(q)$ e quando compare come ultimo incrocio è un sotto-incrocio. Avremo quindi le seguenti sequenze di scambi -// a meno di riordinare gli scambi è facile vedere che $S_i hat(K)(q) = hat(K)(p)$, e che il @lemma-slide-identities si applica alla coppia $hat(K)(p), hat(K)(q)$ per l'indice $i$. Per il @lemma-sum-switches-rotation abbiamo anche che $sum_K (p), sum_K (q)$ sono invarianti a meno di permutazioni cicliche. Possiamo quindi sostituire $lambda(p)$ e $lambda(q)$ con le seguenti sequenze di scambi applicando le giuste rotazioni. + $ + lambda(q) &= (n, n-1, dots, i, dots, 1, 0) \ + lambda(p) &= (n, n-1, dots, i+1, i-1, dots, 1, 0) + $ -// $ -// lambda'(p) &= (i-1, dots, 1, 0, n, n-1, dots, i+1) \ -// lambda'(q) &= (i, i-1, dots, 1, 0, n, n-1, dots, i+1) -// $ + a meno di riordinare gli scambi è facile vedere che $S_i hat(K)(q) = hat(K)(p)$, e che il @lemma-slide-identities si applica alla coppia $hat(K)(p), hat(K)(q)$ per l'indice $i$. Per il @lemma-sum-switches-rotation abbiamo anche che $sum_K (p), sum_K (q)$ sono invarianti a meno di permutazioni cicliche. Possiamo quindi sostituire $lambda(p)$ e $lambda(q)$ con le seguenti sequenze di scambi applicando le giuste rotazioni. -// A questo punto possiamo applicare un argomento simile a quello del punto precedente. + $ + lambda'(p) &= (i-1, dots, 1, 0, n, n-1, dots, i+1) \ + lambda'(q) &= (i, i-1, dots, 1, 0, n, n-1, dots, i+1) + $ -// E questo completa la dimostrazione dell'indipendenza di $Omega_K (p)$ dalla scelta di punto base. -// ] + A questo punto possiamo applicare un argomento simile a quello del punto precedente. -// A questo punto possiamo vedere che $L_K$ verifica gli assiomi della @kauffman-poly-def. + E questo completa la dimostrazione dell'indipendenza di $Omega_K (p)$ dalla scelta di punto base. +] -// #lemma[ -// Sia $i$ un incrocio di un diagramma di link $K$. Allora $L_K$ verifica le identità: +A questo punto possiamo vedere che $L_K$ verifica gli assiomi della @kauffman-poly-def. -// 1. $L[K] + L[S_i K] = z (L[E_i K] + L[e_i K])$ +#lemma[ + Sia $i$ un incrocio di un diagramma di link $K$. Allora $L_K$ verifica le identità: -// 2. $L[#skein.over-twist-medium] = a L [#skein.strand-medium]$, $L[#skein.under-twist-medium] = a^(-1) L [#skein.strand-medium]$ -// ] + 1. $L[K] + L[S_i K] = z (L[E_i K] + L[e_i K])$ -// #proof[ -// Mostriamo per induzione sul numero di incroci nel diagramma. + 2. $L[#skein.over-twist-medium] = a L [#skein.strand-medium]$, $L[#skein.under-twist-medium] = a^(-1) L [#skein.strand-medium]$ +] -// 1. Consideriamo i seguenti casi: #margin-note[ricontrollare tutta questa] +#proof[ + Mostriamo per induzione sul numero di incroci nel diagramma. -// - Se $K$ ha una sola componente ed un incrocio $i$: consideriamo $S_i K$, $E_i K$ e $e_i K$, scegliendo bene $p$ possiamo fare in modo che $i$ sia il primo incrocio nella sequenza di scambi. Se ora consideriamo @kauffman-rec-single-component ovvero + 1. Consideriamo i seguenti casi: #margin-note[ricontrollare tutta questa] -// $ -// L_K (a, z) &colon.eq -// 1 / 2 [ -// sum_(q = p, overline(p)) -// ((-1)^(abs(lambda(q))+1) L[hat(K)(lambda(q))] + z sum_K (lambda(q))) -// ] \ -// &= 1 / 2 [ -// sum_(q = p, overline(p)) -// ((-1)^(n+1) L[hat(K)(q)] + z sum_K (lambda(q))) -// ] -// $ + - Se $K$ ha una sola componente ed un incrocio $i$: consideriamo $S_i K$, $E_i K$ e $e_i K$, scegliendo bene $p$ possiamo fare in modo che $i$ sia il primo incrocio nella sequenza di scambi. Se ora consideriamo @kauffman-rec-single-component ovvero -// A questo punto otteniamo la tesi considerando la differenza nelle espansioni di $Omega_K (p)$ e $Omega_(S_i K) (p)$, in particolare tutti i termini si cancellano tranne il primo: + $ + L_K (a, z) &colon.eq + 1 / 2 [ + sum_(q = p, overline(p)) + ((-1)^(abs(lambda(q))+1) L[hat(K)(lambda(q))] + z sum_K (lambda(q))) + ] \ + &= 1 / 2 [ + sum_(q = p, overline(p)) + ((-1)^(n+1) L[hat(K)(q)] + z sum_K (lambda(q))) + ] + $ -// $ -// => Omega_K (p) + Omega_(S_i K) (p) = z(L[E_i K] + L[e_i K]) \ -// $ + A questo punto otteniamo la tesi considerando la differenza nelle espansioni di $Omega_K (p)$ e $Omega_(S_i K) (p)$, in particolare tutti i termini si cancellano tranne il primo: -// - Se $K$ ha più di una componente e $i$ è un incrocio di una componente con se stessa: in questo caso utilizziamo per induzione @kauffman-rec-multi-component ma osserviamo che $i$ non compare in nessuna sequenza di sollevamento (in quanto le sequenze di sollevamento scambiano solo incroci tra componenti diverse) dunque possiamo applicare l'induzione senza problemi. + $ + => Omega_K (p) + Omega_(S_i K) (p) = z(L[E_i K] + L[e_i K]) \ + $ -// - Se $K$ ha più di una componente e $i$ è un incrocio tra due componenti diverse: possiamo considerare la @kauffman-rec-multi-component e mostrare che ogni addendo verifica l'identità e quindi l'identità seguirà per la loro media: + - Se $K$ ha più di una componente e $i$ è un incrocio di una componente con se stessa: in questo caso utilizziamo per induzione @kauffman-rec-multi-component ma osserviamo che $i$ non compare in nessuna sequenza di sollevamento (in quanto le sequenze di sollevamento scambiano solo incroci tra componenti diverse) dunque possiamo applicare l'induzione senza problemi. -// #{ -// set text(size: small-size) -// $ -// L_K (a, z) colon.eq -// 1 / (2n) [ -// sum_(i=1)^n sum_(q=p_i, overline(p)_i) -// ((-1)^(abs(lambda(q))+1) d kL_(K_i) kL_(K - K_i) + z sum_K (lambda(q))) -// ] -// $ -// } + - Se $K$ ha più di una componente e $i$ è un incrocio tra due componenti diverse: possiamo considerare la @kauffman-rec-multi-component e mostrare che ogni addendo verifica l'identità e quindi l'identità seguirà per la loro media: -// Per quanto riguarda gli addenti che non intersecano nessuna delle due componenti interessate dall'incrocio scelto non ci sono problemi e possiamo applicare l'induzione. Per le componenti che le intersecano ci basta scegliere come prima un punto base in modo appropriato in modo che l'identità sia verificata e procedere come prima. + #{ + set text(size: small-size) + $ + L_K (a, z) colon.eq + 1 / (2n) [ + sum_(i=1)^n sum_(q=p_i, overline(p)_i) + ((-1)^(abs(lambda(q))+1) delta kL_(K_i) kL_(K - K_i) + z sum_K (lambda(q))) + ] + $ + } -// // gli addendi che non intersecano nessuna delle due componenti possiamo procedere per induzione. Altrimenti possiamo scegliere un punto base appropriato e procedere per induzione. + Per quanto riguarda gli addenti che non intersecano nessuna delle due componenti interessate dall'incrocio scelto non ci sono problemi e possiamo applicare l'induzione. Per le componenti che le intersecano ci basta scegliere come prima un punto base in modo appropriato in modo che l'identità sia verificata e procedere come prima. -// 2. La dimostrazione è analoga, basta osservare che possiamo sempre scegliere un punto base che non faccia comparire l'incrocio del ricciolo nella sequenza di scambi. + // gli addendi che non intersecano nessuna delle due componenti possiamo procedere per induzione. Altrimenti possiamo scegliere un punto base appropriato e procedere per induzione. -// E questo conclude la dimostrazione del lemma. -// ] + 2. La dimostrazione è analoga, basta osservare che possiamo sempre scegliere un punto base che non faccia comparire l'incrocio del ricciolo nella sequenza di scambi. -// Come ultimo risultato vediamo che $L_K$ è un invariante di isotopia regolare. + E questo conclude la dimostrazione del lemma. +] -// #lemma[ -// Siano $K, K'$ diagrammi di link equivalenti a meno di isotopia regolare allora $L_K (a, z) = L_(K') (a, z)$. -// ] +Come ultimo risultato vediamo che $L_K$ è un invariante di isotopia regolare. -// #proof[ -// Procediamo sempre per induzione sul numero di incroci e mostriamo l'invarianza per mosse che non aumentano il numero di incroci (l'invarianza per la mossa II che aggiunge due incroci segue come conseguenza dell'induzione#margin-note[da aggiustare]). +#lemma[ + Siano $K, K'$ diagrammi di link equivalenti a meno di isotopia regolare allora $L_K (a, z) = L_(K') (a, z)$. +] -// - Mossa II: Ci sono più casi in base a se la mossa riguarda una o più componenti +#proof[ + Procediamo sempre per induzione sul numero di incroci e mostriamo l'invarianza per mosse che non aumentano il numero di incroci (l'invarianza per la mossa II che aggiunge due incroci segue come conseguenza dell'induzione#margin-note[da aggiustare]). -// - Se la mossa è su una sola componente allora ci basta scegliere il punto base come segue + - Mossa II: Ci sono più casi in base a se la mossa riguarda una o più componenti -// #todo[disegnino] + - Se la mossa è su una sola componente allora ci basta scegliere il punto base come segue -// in questo modo i due incroci non compariranno nella sequenza di scambi. Tutti i termini di ii.a) saranno invarianti per mosse di tipo II e lo sarà anche $L_K$. + #todo[disegnino] -// - Se la mossa riguarda più componenti allora il caso peggiore è il seguente + in questo modo i due incroci non compariranno nella sequenza di scambi. Tutti i termini di ii.a) saranno invarianti per mosse di tipo II e lo sarà anche $L_K$. -// #todo[disegnino] + - Se la mossa riguarda più componenti allora il caso peggiore è il seguente -// Vediamo che $L[K] = L[S_2 S_1 K]$... + #todo[disegnino] -// - Mossa III: + Vediamo che $L[K] = L[S_2 S_1 K]$... -// - Se la mossa è su fili di una sola componente allora possiamo scegliere il punto base sul filo che passa sopra, in questo modo avremo la situazione seguente: + - Mossa III: -// #todo[disegnino] + - Se la mossa è su fili di una sola componente allora possiamo scegliere il punto base sul filo che passa sopra, in questo modo avremo la situazione seguente: -// In un caso possiamo procedere per induzione, se invece ci sono degli splice possiamo utilizzare le seguenti equivalenze di diagrammi e poi procedere per induzione + #todo[disegnino] -// #todo[disegnino] + In un caso possiamo procedere per induzione, se invece ci sono degli splice possiamo utilizzare le seguenti equivalenze di diagrammi e poi procedere per induzione -// - Nel caso in cui la mossa riguardi fili appartenenti a più componenti possiamo utilizzare l'identità della mossa precedente $L[K] = L[S_2 S_1 K]$ per semplificare la situazione e procedere nuovamente per induzione. + #todo[disegnino] -// Questo completa la dimostrazione. -// ] + - Nel caso in cui la mossa riguardi fili appartenenti a più componenti possiamo utilizzare l'identità della mossa precedente $L[K] = L[S_2 S_1 K]$ per semplificare la situazione e procedere nuovamente per induzione. -#todo[work in progress] \ No newline at end of file + Questo completa la dimostrazione. +] \ No newline at end of file diff --git a/src/prelude.typ b/src/prelude.typ index 9265423..7438b1a 100644 --- a/src/prelude.typ +++ b/src/prelude.typ @@ -47,6 +47,11 @@ }) } +#let observation(body) = { + // show figure: statement-style(name, numbered: false) + figure(body, kind: "observation", supplement: [Osservazione], numbering: "1") +} + #let proposition(body, numbered: true) = { // show figure: statement-style(name, numbered: numbered) return figure( diff --git a/src/theme.typ b/src/theme.typ index 4d26bc4..e854fc0 100644 --- a/src/theme.typ +++ b/src/theme.typ @@ -242,6 +242,27 @@ }) } + show figure.where(kind: "observation"): it => { + set align(start) + block({ + strong({ + it.supplement + if it.numbering != none { + [ ] + // current chapter and section + + [#( + ..counter(heading).get().slice(0, 1).map(it => str(it)), + numbering(it.numbering, ..it.counter.at(it.location())), + ).join(".")] + } + [.] + }) + [ ] + it.body + }) + } + show figure.where(kind: "proposition"): it => { set align(start) block({ @@ -308,7 +329,7 @@ show ref: it => { if it.element != none { let el = it.element - if el.func() == figure and (el.kind == "definition" or el.kind == "proposition" or el.kind == "lemma" or el.kind == "theorem") { + if el.func() == figure { link(el.location(), { if it.supplement != auto { it.supplement } else { el.supplement } [ ]

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