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znyLXMR@uDZOkAp)3g1N;P8*ad{8KAzPL>B&j4cJGR^s|}w$UKa9jueJ$R2you;V`% zkvlD(;5U5W6jd4!ta%(R3W21m7%?{&4n*x{tjQzc3k5|BMBNB>t%8iG-)(# zuoyK3H;}=(^rBU0BLPd~ytf2W(H&D0r&vgYjP8!{nccS7&tBNr8o*+rt#;}*`=1ym z49DWExDHyeD<2Wk72UGq?K;r&F@Sti=F4>sOSPLGNLzFGX`k1ordhA0LZrX`dAz&1 zX~C|1zCAs0 bb(R)^3$. +In particolare l'isotopia regolare preserva il framing del nodo. Per quanto riguarda l'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.]. -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 generale 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. Quando invece parliamo di isotopia ambiente ha senso parlare dei link sia come diagrammi che come embedding $bb(S)^1 -> bb(R)^3$. -// 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. +// 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 generale 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. +// 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. -// Citiamo che ci sono lavori riguardanti dare una definizione "intrinseca" del polinomio di Kauffman senza passare da diagrammi. +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. Citiamo che ci sono lavori riguardo il dare una definizione "intrinseca" del polinomio di Kauffman senza passare per i diagrammi. // Questo sarà importante più avanti quando vedremo invarianti di isotopia ambiente definiti passando dall'isotopia regolare. Ad esempio, nel caso del polinomio $F_K$, la definizione si basa sul polinomio $L_K$ che è un invariante per isotopia regolare definito su diagrammi; quindi il calcolo di $F_K$ va fatto fissando un particolare diagramma del link. È tutt'ora un problema aperto dare una definizione di $F_K$ che sia indipendente dalla scelta di un diagramma. +== Diagrammi in forma discendente + +#definition[ + 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 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. In particolare questo induce un ordinamento dall'alto verso il basso delle componenti. + + #figure( + image("assets/standard-unlink-construction.png", width: 12cm), + caption: [ + Un link con tre componenti e punti di partenza direzionati $p_1, p_2, p_3$, con $overline(p) = (p_1, p_2, p_3)$. In $hat(L)(cal(U), overline(p))$ sono evidenziati in #box(radius: 3pt, outset: (y: 3pt), inset: (x: 2pt), fill: rgb("#91cdff"))[azzurro] gli incroci scambiati rispetto al link $L$. + ], + ) +] + +#fact(numbered: true)[ + Un nodo in forma discendente è equivalente ad un nodo banale a meno di isotopia ambiente. +] + +Vediamo ora una proprietà che ci caratterizza i nodi in forma discendente. + +#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 il @std-unknot-trivial-knot, un nodo banale standard è equivalente ad un nodo banale a meno di isotopia ambiente. Quindi possiamo applicare il teorema di Reidemeister ed ottenere una successione di mosse e diagrammi tale che + + $ + K = D_0 stretch(arrow)^(m_1) dots.c stretch(arrow)^(m_n) D_n = #skein.unit-medium + $ + + con $m_i$ mosse di tipo I, II, III. Ora l'idea è modificare questa successione di mosse in modo da ottenere una nuova successione composta solo da mosse II e III che termina con un diagramma composto solo da riccioli. + + Applichiamo induttivamente le seguenti modifiche ad ogni mossa $m_i$ nella sequenza, tenendo traccia di cosa succede man mano agli incroci: + + - Se abbiamo una mossa di tipo I che rimuove un ricciolo, allora scartiamo la mossa in modo da conservare il ricciolo. In particolare, in tutte le mosse successive della sequenza originale, sicuramente non compare questo incrocio poiché questa mossa lo cancellava. Quindi questo sarà uno di quei riccioli che rimarrà nel nuovo diagramma finale. + + - Se abbiamo una mossa di tipo I che aggiunge un ricciolo, allora possiamo introdurlo sfruttando l'@obs-whitney-trick ovvero il trucco di Whitney. Possiamo creare due riccioli utilizzando solo mosse II e III in modo tale che uno dei due sia orientato allo stesso modo di quello introdotto dal ricciolo della sequenza originale, l'altro sarà in più ma possiamo ignorarlo come nel caso precedente. + + #figure(image("assets/modified-curl-add-before.png", width: 6.5cm)) + #figure(image("assets/modified-curl-add-after.png", width: 13cm)) + + Abbiamo aggiunto così due incroci uno dei quali corrisponde ad uno della vecchia sequenza che verrà cancellato dalle mosse originali mentre l'altro è uno di quei riccioli che rimarranno nel diagramma finale. + + - Se abbiamo una mossa di tipo II o III, allora questa mossa interesserà una certa regione del diagramma. Per prima cosa utilizziamo l'@obs-curls-pass-through per spostare tutti quei riccioli che si trovano nella regione in questione al di fuori di essa e poi applichiamo la mossa originale. + + #figure(image("assets/modified-r2-before.png", width: 6.5cm)) + #figure( + image("assets/modified-r2-after.png", width: 8.5cm), + caption: [Esempio per la mossa II che rimuove un incrocio], + ) + + In questo caso abbiamo solo spostato dei riccioli ed applicato la stessa mossa della sequenza originale quindi non abbiamo introdotto incroci in più rispetto alla sequenza originale. + + Otteniamo così una nuova successione di diagrammi $D'_i$ e di mosse $m_i^*$ solo di tipo II o III: + + $ + K = D'_0 stretch(arrow)^(m_1^*) dots.c stretch(arrow)^(m_n^*) D'_n + $ + + Vorremo ora vedere che il diagramma $D'_n$ è composto solo da riccioli. Questo segue dal fatto che nella successione originale tutti gli incroci si cancellavano, ciò accade anche nella nuova successione tranne per gli incroci che abbiamo aggiunto modificando le mosse di tipo I, ma tutti questi sono dei riccioli quindi $D'_n$ sarà composto solo da riccioli. +] + +*Corollario.* Un'altro modo di vedere il lemma precedente è che il diagramma di un nodo in forma discendente è equivalente a meno di isotopia regolare ad uno composto solo da riccioli. + == Writhe Vediamo ora un primo invariante di isotopia regolare per link orientati, per prima cosa definiamo il segno di un incrocio. @@ -605,19 +673,19 @@ In particolare abbiamo le seguenti identità per i polinomi $L_K$ e $F_K$ in rel Prima di descrivere la definizione induttiva diamo alcune definizioni per certe manipolazioni dei diagrammi di nodi e link. #definition[ - 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 $cal(U)$. + Sia $K$ un diagramma di un nodo, $cal(U)$ la sua ombra planare 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 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. +// - Ricordiamo la definizione di *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. In particolare questo induce un ordinamento dall'alto verso il basso delle componenti. +// 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), - caption: [ - Un link con tre componenti e punti di partenza direzionati $p_1, p_2, p_3$, con $overline(p) = (p_1, p_2, p_3)$. In $hat(L)(cal(U), overline(p))$ sono evidenziati in #box(radius: 3pt, outset: (y: 3pt), inset: (x: 2pt), fill: rgb("#91cdff"))[azzurro] gli incroci scambiati rispetto al link $L$. - ], - ) +// #figure( +// image("assets/standard-unlink-construction.png", width: 12cm), +// caption: [ +// Un link con tre componenti e punti di partenza direzionati $p_1, p_2, p_3$, con $overline(p) = (p_1, p_2, p_3)$. In $hat(L)(cal(U), overline(p))$ sono evidenziati in #box(radius: 3pt, outset: (y: 3pt), inset: (x: 2pt), fill: rgb("#91cdff"))[azzurro] gli incroci scambiati rispetto al link $L$. +// ], +// ) - 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 @@ -676,10 +744,6 @@ Sia $lambda = (lambda_n, dots, lambda_0)$ una sequenza di etichette di incroci d dove $abs(lambda) colon.eq n$. -#fact(numbered: true)[ - 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 $hat(K)(cal(U), p) = S_lambda_n dots.c space S_lambda_0 K$. #observation[ @@ -696,66 +760,17 @@ Sia $lambda = (lambda_n, dots, lambda_0)$ una sequenza di etichette di incroci d #definition[ Sia $D$ il diagramma di un link $L$, allora una *funzione di slacciamento* è una funzione $h$ sui punti di un diagramma a valori in $RR$, con due valori in corrispondenza degli incroci, che corrisponde ad una funzione $h : L -> RR$ con le seguenti proprietà: - 1. Se una componente $C_i$ sovrasta una componente $C_j$ allora #h(1fr) - - $ - forall x_i in C_i, x_j in C_j quad h(C_i) > h(C_j) - $ + 1. Se una componente $C_i$ sovrasta una componente $C_j$ allora $forall x_i in C_i$ e $forall x_j in C_j$, $h(C_i) > h(C_j)$. - 2. ... + 2. Su ogni componente $C_i$, la funzione $h$ è strettamente monotona crescente da $b_i in C_i$ a $t_i in C_i$ in entrambe le direzioni attorno a $C_i$. - 3. ... + 3. Ad un incrocio $x$, avrà due valori $h(x) = (h_+, h_-)$ rispettivamente per il caso in cui passa da sopra e da sotto e vale $h_+ > h_-$. ] ] -#align(center, line(stroke: 0.5pt, length: 50%)) - -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 il @std-unknot-trivial-knot, un nodo banale standard è equivalente ad un nodo banale a meno di isotopia ambiente. Quindi possiamo applicare il teorema di Reidemeister ed ottenere una successione di mosse e diagrammi tale che - - $ - K = D_0 stretch(arrow)^(m_1) dots.c stretch(arrow)^(m_n) D_n = #skein.unit-medium - $ - - con $m_i$ mosse di tipo I, II, III. Ora l'idea è modificare questa successione di mosse in modo da ottenere una nuova successione composta solo da mosse II e III che termina con un diagramma composto solo da riccioli. - - Applichiamo induttivamente le seguenti modifiche ad ogni mossa $m_i$ nella sequenza, tenendo traccia di cosa succede man mano agli incroci: - - - Se abbiamo una mossa di tipo I che rimuove un ricciolo, allora scartiamo la mossa in modo da conservare il ricciolo. In particolare, in tutte le mosse successive della sequenza originale, sicuramente non compare questo incrocio poiché questa mossa lo cancellava. Quindi questo sarà uno di quei riccioli che rimarrà nel nuovo diagramma finale. - - - Se abbiamo una mossa di tipo I che aggiunge un ricciolo, allora possiamo introdurlo sfruttando l'@obs-whitney-trick ovvero il trucco di Whitney. Possiamo creare due riccioli utilizzando solo mosse II e III in modo tale che uno dei due sia orientato allo stesso modo di quello introdotto dal ricciolo della sequenza originale, l'altro sarà in più ma possiamo ignorarlo come nel caso precedente. +// #align(center, line(stroke: 0.5pt, length: 50%)) - #figure(image("assets/modified-curl-add-before.png", width: 6.5cm)) - #figure(image("assets/modified-curl-add-after.png", width: 13cm)) - - Abbiamo aggiunto così due incroci uno dei quali corrisponde ad uno della vecchia sequenza che verrà cancellato dalle mosse originali mentre l'altro è uno di quei riccioli che rimarranno nel diagramma finale. - - - Se abbiamo una mossa di tipo II o III, allora questa mossa interesserà una certa regione del diagramma. Per prima cosa utilizziamo l'@obs-curls-pass-through per spostare tutti quei riccioli che si trovano nella regione in questione al di fuori di essa e poi applichiamo la mossa originale. - - #figure(image("assets/modified-r2-before.png", width: 6.5cm)) - #figure( - image("assets/modified-r2-after.png", width: 8.5cm), - caption: [Esempio per la mossa II che rimuove un incrocio], - ) - - In questo caso abbiamo solo spostato dei riccioli ed applicato la stessa mossa della sequenza originale quindi non abbiamo introdotto incroci in più rispetto alla sequenza originale. - - Otteniamo così una nuova successione di diagrammi $D'_i$ e di mosse $m_i^*$ solo di tipo II o III: - - $ - K = D'_0 stretch(arrow)^(m_1^*) dots.c stretch(arrow)^(m_n^*) D'_n - $ - - Vorremo ora vedere che il diagramma $D'_n$ è composto solo da riccioli. Questo segue dal fatto che nella successione originale tutti gli incroci si cancellavano, ciò accade anche nella nuova successione tranne per gli incroci che abbiamo aggiunto modificando le mosse di tipo I, ma tutti questi sono dei riccioli quindi $D'_n$ sarà composto solo da riccioli. -] - -*Corollario.* Un'altro modo di vedere il lemma precedente è che il diagramma di un nodo in forma discendente è equivalente a meno di isotopia regolare ad uno composto solo da riccioli. +// Enunciamo ora un lemma che utilizzeremo più volte in seguito. #align(center, line(stroke: 0.5pt, length: 50%)) @@ -865,7 +880,7 @@ sempre con $delta = (a + a^(-1)) slash z - 1$. // la definizione è sotto perché gli enumerate non vanno d'accordo con le figure -1. Se $K = hat(K)(cal(U), p)$ è un _nodo banale standard_ per un qualche $p$ allora +1. #marker[i)] Se $K = hat(K)(cal(U), p)$ è un _nodo banale standard_ per un qualche $p$ allora $ kL_K (a, z) colon.eq a^w(K) @@ -1122,21 +1137,51 @@ Ora rimane da verificare l'invarianza per scelta di punto base per il caso @kauf $ L_K (a, z) colon.eq - 1 / 2 [ + 1 / 2 sum_(q = p, overline(p)) ((-1)^(abs(lambda(q))+1) L[hat(K)(lambda(q))] + z sum_K (lambda(q))) - ] $ La dimostrazione procede sempre mostrando l'indipendenza dalla scelta di punto base. Vediamo prima alcune proprietà dei nodi banali standard: +// Per come abbiamo definito $L_K$ abbiamo che, dato $p$ e dati $i$ e $j$ indici dei primi incroci nelle sequenze di scambi rispetto a $p$ e $overline(p)$: + +// $ +// L[K] = 1/2 [ +// & z ( L[E_(i) K] + L[e_(i) K]) - S_(i) K + \ +// + & z ( L[E_j K] + L[e_j K]) - S_j K +// ] +// $ + +// Questo segue semplicemente sommando le espressioni per $L[K]$ e $L[S_i K]$. + #diff-add[ + Dimostriamo ora il seguente lemma nel caso più semplice di aver definito $L_K$ rispetto ad una sola orientazione ovvero, dato $p$ punto base e $0$ indice del primo incrocio nella sequenza di scambi indotta da $p$: + + $ + L_K (a, z) & colon.eq + (-1)^(abs(lambda(p))+1) L[hat(K)(lambda(p))] + z sum_K (lambda(p)) \ + & = z (L[E_0 K] + L[e_0 K]) - L[S_0 K] + $ + + l'ultima uguaglianza segue semplicemente considerando $L[K]$ e $L[S_0 K]$ e sottraendo membro a membro. + #lemma[ - Se $K$ ammette una funzione di slacciamento allora $L_K = a^w(K)$. + Se $K$ è un diagramma di link e ammette una funzione di slacciamento allora $L_K = a^w(K)$. ] #proof[ - #todo[work in progress] + Se i punti a valle in $K$ precedono immediatamente i rispettivi punti base allora i diagrammi sono in forma discendente e possiamo utilizzare direttamente la @kauffman-rec-inductive-def di $L_K$ nel caso @kauffman-rec-i. Procediamo per sotto-induzione sul numero totale di incroci e sulla distanza dei punti a valle dai punti base. + + Per prima cosa osserviamo che possiamo ignorare gli incroci tra componenti diverse. L'esistenza di una funzione di slacciamento ci induce un ordinamento delle componenti e per il teorema della curva di Jordan tutti i segni degli incroci tra componenti diverse si cancellano. + + Consideriamo ora una singola componente, sia $i$ il primo incrocio dopo il punto a valle $b$. + + - Se partendo da $b$, l'incrocio $i$ è un sotto-incrocio, $h$ può essere modificata + ] + + #fact[ + L'enunciato precedente si generalizza anche con la nostra definizione di $L_K$ che prende in considerazione entrambe le direzioni. La dimostrazione è analoga ma serve tenere conto di come si presentano gli incroci in entrambe le direzioni. ] ]