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Ci sono due casi, in uno otteniamo _due nodi banali in forma standard_, nell'altro _un solo nodo banale_ (non necessariamente in forma discendente). - 2. #diff-add[ - Il caso di splice con una sola componente ammette una funzione di slacciamento. - ] + 2. Il caso di splice con una sola componente ammette una funzione di slacciamento. // #diff-del[La proprietà del @std-unknot-to-curls, ovvero che i nodi banali standard sono equivalenti a meno di isotopia regolare a diagrammi di nodi formati solo da riccioli, si estende al diagramma del caso dello splice con una sola componente.] @@ -1213,19 +1211,34 @@ La dimostrazione procede sempre mostrando l'indipendenza dalla scelta di punto b #proof[ Per la definizione di nodo banale standard, la prima occorrenza di $i$ è un sopra-incrocio e osserviamo che la parte di diagramma percorsa tra le due occorrenze di $i$ sarà sicuramente sopra il resto del diagramma (la parte che rimane è quella dopo la seconda occorrenza di $i$). + #figure(image("assets/splice-circuits.png", width: 4cm)) + Dunque abbiamo due casi, in una abbiamo due componenti una sopra l'altra, ognuna delle quali sarà ancora in forma discendente in quanto eredita la proprietà secondo cui la prima volta che si passa su un incrocio questo è un sopra-incrocio. - Se assumiamo che il diagramma del caso precedente sia della forma $E_i hat(K) = hat(K)_1 union hat(K)_2$ con $hat(K)_1$, $hat(K)_2$ i due nodi banali standard con $hat(K)_1$ sovrastante $hat(K)_2$, allora l'altro caso sarà $e_i hat(K) = hat(K)_1 hash hat(K)_2^*$ dove $hat(K)_2^*$ è ottenuto percorrendo $hat(K)_2$ in direzione opposta. + Se assumiamo che il diagramma del caso precedente sia della forma $E_i hat(K) = hat(K)_1 union hat(K)_2$ con $hat(K)_1$, $hat(K)_2$ i due nodi banali standard con $hat(K)_1$ sovrastante $hat(K)_2$, allora l'altro caso sarà $e_i hat(K) = hat(K)_1 hash hat(K)_2^*$ dove $hat(K)_2^*$ è ottenuto percorrendo $hat(K)_2$ in direzione opposta. In questo caso $e_i hat(K)$ non è in forma discendente ma ammette una funzione di slacciamento costruita come segue. - #figure(image("assets/splice-circuits.png", width: 4cm)) - - #diff-del[ - Usiamo ora il @std-unknot-to-curls per cui un nodo banale standard è equivalente a meno di isotopia regolare al diagramma di un nodo formato solo da riccioli. + #figure( + // + grid(gutter: 0.5em, image("assets/splices-with-untying-function.png", width: 11cm), grid( + // + columns: (1.5em, 1fr, 1fr, 1fr, 2em), + [], + $hat(K)$, + $E_i hat(K)$, + $E_i hat(K)$, + [], + )), + ) - Siano $hat(K)'_1$ e $hat(K)'_2$ i diagrammi a cui sono equivalenti rispettivamente $hat(K)_1$ e $hat(K)_2$. Ora possiamo applicare le mosse che semplificano $hat(K)_1$ indipendentemente dalle mosse che semplificano $hat(K)_2$, in quanto uno sovrasta l'altro e possiamo spostare un diagramma sopra un altro utilizzando solo mosse di tipo II o III. A questo punto avremo ridotto i diagrammi ad uno della forma $hat(K)'_1 hash hat(K)'_2$ che sarà equivalente a $hat(K)_1 hash hat(K)_2^*$ e composto solo da riccioli. - ] + Osserviamo che se effettuiamo uno splice al primo incrocio dopo il punto base, passeremo prima dalla componente $K_2^*$ in verso opposto, questa parte sarà quindi in forma ascendente. Al secondo incontro dello splice torneremo sulla componente $K_1$ che percorriamo nel verso originale quindi sarà in forma discendente. Ora notiamo che $K_2$ è la parte tra la prima occorrenza di $i$ e la seconda quindi sarà sicuramente sopra tutto il resto del diagramma. Per concludere come funzione di slacciamento possiamo prenderne una che ha come punto in cima $t$ e come punto a valle $b$ come mostrato in figura. I colori indicano, in riferimento alle orientazioni, quelle parti che sono rispettivamente #box(radius: 3pt, outset: (y: 3pt), inset: (x: 2pt), fill: green.mix(white))[ascendenti] o #box(radius: 3pt, outset: (y: 3pt), inset: (x: 2pt), fill: red.mix(white))[discendenti]. ] +// #diff-del[ +// Usiamo ora il @std-unknot-to-curls per cui un nodo banale standard è equivalente a meno di isotopia regolare al diagramma di un nodo formato solo da riccioli. + +// Siano $hat(K)'_1$ e $hat(K)'_2$ i diagrammi a cui sono equivalenti rispettivamente $hat(K)_1$ e $hat(K)_2$. Ora possiamo applicare le mosse che semplificano $hat(K)_1$ indipendentemente dalle mosse che semplificano $hat(K)_2$, in quanto uno sovrasta l'altro e possiamo spostare un diagramma sopra un altro utilizzando solo mosse di tipo II o III. A questo punto avremo ridotto i diagrammi ad uno della forma $hat(K)'_1 hash hat(K)'_2$ che sarà equivalente a $hat(K)_1 hash hat(K)_2^*$ e composto solo da riccioli. +// ] + // #grid( // columns: (1fr, auto), // gutter: 1em, @@ -1260,16 +1273,16 @@ La dimostrazione procede sempre mostrando l'indipendenza dalla scelta di punto b // #todo[work in progress] // ] -#diff-del[ - *Corollario.* Poiché $e_i hat(K)$ è equivalente a meno di isotopia regolare ad un diagramma formato solo da riccioli, possiamo usare il caso @kauffman-inductive-hyp-c dell'ipotesi induttiva su $L_K$ che ci dà l'invarianza per isotopia regolare di $L_K$. Unito al fatto che il writhe è un invariante di isotopia regolare otteniamo che $L[e_i hat(K)] = a^w(e_i hat(K))$. -] -#diff-add[ - *Corollario.* Sia $K$ un diagramma di un nodo e $e_i hat(K)$ il caso di splice che forma una sola componente. Per il lemma precedente $e_i hat(K)$ ammette una funzione di slacciamento,dunque possiamo applicare il caso @kauffman-inductive-hyp-d dell'ipotesi induttiva della @kauffman-inductive-hyp-def su $L_K$ ed ottenere che +// #diff-del[ +// *Corollario.* Poiché $e_i hat(K)$ è equivalente a meno di isotopia regolare ad un diagramma formato solo da riccioli, possiamo usare il caso @kauffman-inductive-hyp-c dell'ipotesi induttiva su $L_K$ che ci dà l'invarianza per isotopia regolare di $L_K$. Unito al fatto che il writhe è un invariante di isotopia regolare otteniamo che $L[e_i hat(K)] = a^w(e_i hat(K))$. +// ] +// #diff-add[ +*Corollario.* Sia $K$ un diagramma di un nodo e $e_i hat(K)$ il caso di splice che forma una sola componente. Per il lemma precedente $e_i hat(K)$ ammette una funzione di slacciamento, dunque possiamo applicare il caso @kauffman-inductive-hyp-d dell'ipotesi induttiva della @kauffman-inductive-hyp-def su $L_K$ ed ottenere che - $ - L[e_i hat(K)] = a^w(e_i hat(K)) - $ -] +$ + L[e_i hat(K)] = a^w(e_i hat(K)) +$ +// ] // Utilizzando questi ultimi risultati identità per nodi banali standard che ci mostrano cosa succede quando facciamo avanzare il punto base di un incrocio. @@ -1441,7 +1454,7 @@ Possiamo ora mostrare l'invarianza per scelta di punto base di $Omega_K (p)$. $ 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) + // hat(K)(q) &= S_n space.med hat(K)(p) $ inoltre per il @lemma-slide-identities abbiamo anche che @@ -1508,7 +1521,7 @@ Possiamo ora mostrare l'invarianza per scelta di punto base di $Omega_K (p)$. Dunque riordinando gli scambi ed applicando $S_i$ per ultimo, possiamo notare che $S_i hat(K)(q) = hat(K)(p)$. - Con un argomento simile al @lemma-slide-identities, possiamo applicare l'enunciato alla coppia $hat(K)(p), hat(K)(q)$ per l'indice $i$ anche se in questo caso l'indice $i$ è il primo _subito prima_ di $q$. Inoltre per il @lemma-sum-switches-rotation abbiamo 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 all'indietro. + Con un argomento simile a quello del @lemma-slide-identities, possiamo applicare l'enunciato alla coppia $hat(K)(p), hat(K)(q)$ per l'indice $i$ anche se in questo caso l'indice $i$ è il primo _subito prima_ di $q$. Inoltre per il @lemma-sum-switches-rotation abbiamo 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 all'indietro. $ lambda'(p) &= (i-1, dots, 1, 0, n, n-1, dots, i+1) \ @@ -1539,14 +1552,21 @@ A questo punto possiamo vedere che $L_K$ verifica gli assiomi della @kauffman-po $ 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))) - ] \ - &= 1 / 2 [ + \ + &= 1 / 2 sum_(q = p, overline(p)) ((-1)^(n+1) L[hat(K)(q)] + z sum_K (lambda(q))) - ] + $ + + Ricordiamo le definizioni dei termini $Omega_K (p)$ e $sum_K (lambda(q))$: + + $ + sum_K (lambda(q)) & colon.eq sum_(i=0)^n (-1)^i (L[A_i^lambda K] + L[B_i^lambda K]) \ + + Omega_K (q) & colon.eq (-1)^(abs(lambda) + 1) L[hat(K)(lambda)] + z sum_K (lambda) $ La tesi segue 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: