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Esisterà quindi una sequenza di mosse di tipi I, II, III che ci porta al diagramma $#skein.unit$. Rimuovendo da questa successione tutte le mosse di tipo I e modificando 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. +] -Questa osservazione ci permette di calcolare esplicitamente $L_(hat(K)(cal(U), p))$ che può essere risolto usando solo gli assiomi #link()[ii.b), ii.c), ii.d)] ovvero quelli per $#skein.over-twist-medium$, $#skein.under-twist-medium$, $#skein.unit-medium$. Segue facilmente per induzione che $L_hat(K) (a, z) = a^w(K)$. +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 induttivo +=== Caso base: nodo banale standard + +Utilizzando il lemma precedente possiamo mostrare che calcolare esplicitamente $L_(hat(K)(cal(U), p))$ che può essere risolto usando solo gli assiomi #link()[ii.b), ii.c), ii.d)] ovvero quelli per $#skein.over-twist-medium$, $#skein.under-twist-medium$, $#skein.unit-medium$. Segue facilmente per induzione che $L_hat(K) (a, z) = a^w(K)$. + +=== Caso ricorsivo L'idea per la definizione induttiva è di considerare dato un nodo o un link $K$ il suo nodo o link discendente standard $hat(K)$. Questo induce una sequenza di indici $lambda$ che indichiamo con $0, dots, n$. Se applichiamo incrementalmente gli scambi a $K$, ovvero consideriamo $S_i dotss S_0 K$, ed applichiamo la relazione skein principale a questi diagrammi otteniamo le seguenti relazioni @@ -881,19 +891,19 @@ Nel corso della dimostrazione tutti gli argomenti per induzione si baseranno sul Calcoliamo ora $sum_K (lambda) - sum_K (mu)$, per prima cosa allineiamo i termini come segue: #{ - set text(size: footnote-size) + set text(size: small-size) $ - & (L[E_0 K] + L[e_0 K]) + \ - & - ( L[E_1 S_0 K] + L[e_1 S_0 K]) + & #h(2em) & (L[E_1 K] + L[e_1 K]) + \ - & + ( L[E_2 S_1 S_0 K] + L[e_2 S_1 S_0 K]) + & & - ( L[E_2 S_1 K] + L[e_2 S_1 K]) + \ - & space dots.v & & + ( L[E_3 S_2 S_1 K] + L[e_3 S_2 S_1 K]) + \ - & + (-1)^n ( L[E_n S_(n-1) dotss S_0 K] + L[e_n S_(n-1) dotss S_0 K] ) & & space dots.v \ - & & & + (-1)^n ( L[E_0 S_n dotss S_1 K] + L[e_0 S_n dotss S_1 K] ) \ + & & (L[E_0 K] + L[e_0 K]) + #h(2em) & & & \ + & & - ( L[E_1 S_0 K] + L[e_1 S_0 K]) + #h(2em) & & & (L[E_1 K] + L[e_1 K]) + \ + & & + ( L[E_2 S_1 S_0 K] + L[e_2 S_1 S_0 K]) + #h(2em) & & & - ( L[E_2 S_1 K] + L[e_2 S_1 K]) + \ + & & space dots.v space #h(2em) & & & + ( L[E_3 S_2 S_1 K] + L[e_3 S_2 S_1 K]) + \ + & & + (-1)^n ( L[E_n S_(n-1) dotss S_0 K] + L[e_n S_(n-1) dotss S_0 K] ) #h(2em) & & & space dots.v \ + & & #h(2em) & & & + (-1)^n ( L[E_0 S_n dotss S_1 K] + L[e_0 S_n dotss S_1 K] ) \ $ } - a questo punto raggruppiamoli fattorizzando per righe: + A questo punto raggruppiamoli fattorizzando per righe: $ sum_K (lambda) - sum_K (mu) = + ( & L[E_0 K] + L[e_0 K]) + \ @@ -904,9 +914,17 @@ 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] ) $ - #let blank = $#h(0.125em) #rect(width: 0.8em * 1.41, height: 0.8em, stroke: 0.5pt)#h(0.125em)$ + #let blank = { + $#h(0.125em) #{ + rect(width: 0.8em * 1.41, height: 0.8em, stroke: 0.5pt, place(dx: -0.2em, dy: -0.55em, { + set text(size: 10pt) + + $dots$ + })) + } #h(0.125em)$ + } - permutando i termini di tutte le righe tranne la prima e l'ultima possiamo riscriverli in modo da averli nella forma "$L[#blank K] + L[#blank S_0 K]$" + Permutando i termini di tutte le righe tranne la prima e l'ultima possiamo riscriverli in modo da averli nella forma $L[#blank K] + L[#blank S_0 K]$: $ sum_K (lambda) - sum_K (mu) = @@ -918,7 +936,7 @@ 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] ) $ - ora utilizziamo un'osservazione precedente riguardo l'ordine di applicazione delle operazioni $S_i$ quindi otteniamo $S_(n-1) dotss S_1 S_0 K = S_0 S_(n-1) dotss S_1 K$, dunque possiamo applicare l'induzione ovunque troviamo + Ora utilizziamo un'osservazione precedente riguardo l'ordine di applicazione delle operazioni $S_i$ quindi otteniamo $S_(n-1) dotss S_1 S_0 K = S_0 S_(n-1) dotss S_1 K$, dunque possiamo applicare l'induzione ovunque troviamo $ L[#blank] + L[S_i #blank] = z (L [E_i #blank] + L[e_i #blank]) @@ -960,14 +978,14 @@ 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 $E_0 K$ e $e_0 K$, dunque possiamo riscrivere l'espressione precedente come + Infine osserviamo che i termini in $z(#blank)$ sono proprio le somme incrementali sugli scambi per $E_0 K$ e $e_0 K$, 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)) $ - 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 $gamma = (n, dots, 1)$ e osservando che con la convenzione di Kauffman $abs(gamma) + 1 = (n - 1) + 1 = n$, otteniamo che $ Omega_K(gamma) @@ -988,7 +1006,7 @@ Nel corso della dimostrazione tutti gli argomenti per induzione si baseranno sul *Corollario.* Nella definizione della forma chiusa di $L_K$, il caso ii.a) non dipende dalla scelta del punto base. #proof[ - Rivediamo l'espressione in questione + Rivediamo l'espressione in questione, siano $p_i$ dei punti base direzionati sulle componenti di $K$ $ L_K (a, z) colon.eq @@ -1001,4 +1019,80 @@ Nel corso della dimostrazione tutti gli argomenti per induzione si baseranno sul se applichiamo una $1$-rotazione alle sequenze di scambi le lunghezze non cambiano ed abbiamo appena mostrato che $sum_K (lambda(q)))$ è invariante per $1$-rotazioni. A questo punto possiamo estendere l'invarianza per $1$-rotazioni a tutte le possibili permutazioni cicliche delle sequenze di scambi 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 $K$. + +$ + 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). +] + +#proof[ + #todo[work in progress] +] + +#lemma[ + La proprietà del lemma @std-unknot-to-curls si estende al diagramma $e_i hat(K)$. +] + +#proof[ + #todo[work in progress] +] + +#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. $S_i hat(K) = hat(K)'$ + + 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. + + 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. + + Supponiamo che $w(e_i hat(K)) = w$, allora in quanto abbiamo solo rimosso un incrocio necessariamente + + $ + {w(hat(K)), w(S_i hat(K))} = {w-1, w+1} + $ + + ovvero i writhe di $hat(K)$ e $S_i hat(K)$ differiscono di $2$, assumiamo di essere nel caso $w(hat(K)) = w - 1$. + + 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 $w_1 + w_2 = w$. + + $ + 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))/z - a^w + $ + + che dunque verifica l'identità: + + $ + 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)]) \ + $ + + E questo conclude la dimostrazione delle due proprietà. +] + +#lemma[ + Sia $K$ un diagramma di un nodo. Allora ii.b) non dipende dalla scelta di punto base e più precisamente vale + + $ + Omega_K(p) = (-1)^(abs(lambda(p)) + 1) + $ ] \ No newline at end of file