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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 orientati $K$ come + Definiamo $F_K in Z[a, a^(-1), z, z^(-1)]$ per diagrammi di link orientati $K$ come $ F_K colon.eq a^(-w(K)) kL_K $ @@ -464,7 +464,7 @@ Come già anticipato in precedenza, vedremo che questo è un invariante di isoto => kL[#skein.over] + kL[#skein.under] = z (kL[#skein.h] + kL[#skein.v]) $ - quindi passare al mirror ha l'effetto di scambiare solamente $a$ con $1 slash a$ ma il resto della valutazione rimane identico. + quindi passare al mirror ha l'effetto di scambiare solamente $a$ con $1 slash a$ ma il resto della valutazione rimane identica. Infine per $F_m(K)$ basta osservare che $w(m(K)) = -w(K)$ per una motivazione analoga. ] @@ -497,7 +497,9 @@ $ => delta & = (a + a^(-1)) slash z - 1 $ -Questo valore $delta = (a + 1 slash a) slash z - 1$ ricomparirà anche in seguito ed è il coefficiente che il polinomio di Kauffman introduce quando abbiamo più componenti disgiunte. Possiamo anche trovare il risultato per il link di Hopf come segue +Questo valore $delta = (a + 1 slash a) slash z - 1$ ricomparirà anche in seguito ed è il coefficiente che il polinomio di Kauffman introduce quando abbiamo più componenti disgiunte (ovvero $L[K_1 union.sq K_2] = delta L[K_1] L[K_2]$). + +Possiamo anche trovare il valore per il link di Hopf come segue // #figure(image("assets/implicit-calc-2.png")) @@ -524,7 +526,7 @@ $ - (a + a^(-1)) z^(-1) + 1 + (a + a^(-1)) z $ -E del nodo trifoglio +E ad esempio anche del nodo trifoglio: // #figure(image("assets/implicit-calc-3.png")) @@ -552,7 +554,7 @@ $ $ #definition[ - Diremo che un diagramma è _split_ se è l'unione disgiunta di due diagrammi separati, in questo caso scriviamo $K = K_1 union.sq K_2$ dove $K_1$ e $K_2$ sono le due componenti disgiunte. + Diremo che un diagramma è *split* se è l'unione disgiunta di due diagrammi separati, in questo caso scriviamo $K = K_1 union.sq K_2$ dove $K_1$ e $K_2$ sono le due componenti disgiunte. ] #definition[ @@ -581,13 +583,13 @@ $ == Strumenti Forma Induttiva -Prima di passare a descrivere la definizione induttiva diamo alcune definizioni di manipolazioni di un diagramma. +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 $U colon.eq pi(K) subset bb(R)^2$] 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#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* associato 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 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. Questa definizione si estende anche a diagrammi di link prendendo un insieme di punti base ordinati come mostrato in figura. @@ -633,16 +635,14 @@ Prima di passare a descrivere la definizione induttiva diamo alcune definizioni 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. -#pagebreak() - -Sia ora $lambda = (lambda_n, dots, lambda_0)$ una sequenza di etichette di suoi incroci. +Sia $lambda = (lambda_n, dots, lambda_0)$ una sequenza di etichette di suoi incroci. - Definiamo le seguenti operazioni compatte per una sequenza di scambi seguita da uno splice finale $ - A_i^lambda = E_lambda_i S_lambda_(i-1) dots.c space S_lambda_0 + A_i^lambda colon.eq E_lambda_i S_lambda_(i-1) dots.c space S_lambda_0 #h(2em) - B_i^lambda = e_lambda_i S_lambda_(i-1) dots.c space S_lambda_0 + 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. @@ -655,12 +655,14 @@ Sia ora $lambda = (lambda_n, dots, lambda_0)$ una sequenza di etichette di suoi Omega_K (lambda) & colon.eq (-1)^(abs(lambda) + 1) L[hat(K)(lambda)] + z sum_K (lambda) $ - dove $abs(lambda) = n$ per $lambda$ come sopra. + dove $abs(lambda) colon.eq n$ per $lambda$. + +*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. #definition[ - Dati $K_1, K_2$ diagrammi di link, diciamo che $K_1$ *sovrasta* $K_2$ se per 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 tra $K_1$ e $K_2$ abbiamo che sono tutti sopra-incroci. ] == Forma Induttiva @@ -668,31 +670,33 @@ Sia ora $lambda = (lambda_n, dots, lambda_0)$ una sequenza di etichette di suoi Per prima cosa enunciamo un lemma che utilizzeremo più volte in seguito. #lemma[ - Uno nodo banale standard o è composto interamente da riccioli o ammette una sequenza di mosse di tipo II e III che portano ad un nodo formato solo da riccioli. + 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 il diagramma di un nodo non orientato $K$ allora, preso $p$ un qualsiasi punto di partenza direzionato, $hat(K)(cal(U), p)$ sarà un nodo banale. 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. + 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. ] 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 -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 $L_hat(K) (a, z) = a^w(K)$. +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 -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 +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 $p$. + +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: $ L[K] + L[S_0 K] &= z( L[E_0 K] + L[e_0 K] ) \ - L[S_0 K] + L[S_1 S_0 K] &= z( L[E_0 S_0 K] + L[e_0 S_0 K] ) \ + L[S_0 K] + L[S_1 S_0 K] &= z( L[E_1 S_0 K] + L[e_1 S_0 K] ) \ & space dots.v \ L[S_(n-1) dotss S_0 K] + L lr([underbrace(S_n dotss S_0 K, hat(K))], size: #1.125em) &= z (L[E_n S_(n-1) dotss S_0 K] + L[e_n S_(n-1) dotss S_0 K]) $ -se ora sommiamo e sottraiamo membro a membro queste equazioni otteniamo la seguente identità +Ora, sommando e sottraendo membro a membro queste equazioni otteniamo la seguente identità #context { set align(center) @@ -707,7 +711,7 @@ se ora sommiamo e sottraiamo membro a membro queste equazioni otteniamo la segue $z( L[E_0 K] + L[e_0 K] )$, $-(L[S_0 K] + L[S_1 S_0 K])$, [], - $-z( L[E_0 S_0 K] + L[e_0 S_0 K] )$, + $-z( L[E_1 S_0 K] + L[e_1 S_0 K] )$, $dots.v$, $=$, $dots.v$, @@ -732,7 +736,7 @@ notiamo che possiamo cancellare tutti i termini del membro di sinistra che compa $z( L[E_0 K] + L[e_0 K] )$, $-(cancel(L[S_0 K]) + cancel(L[S_1 S_0 K]))$, [], - $-z( L[E_0 S_0 K] + L[e_0 S_0 K] )$, + $-z( L[E_1 S_0 K] + L[e_1 S_0 K] )$, $dots.v$, $=$, $dots.v$, @@ -751,11 +755,11 @@ Da cui otteniamo un'espressione ricorsiva per $L_K$: => L_K & = (-1)^(n+1) L_(hat(K)) + z sum_(i=0)^n (-1)^i ( - L[E_n S_(n-1) dotss S_0 K] + L[e_n S_(n-1) dotss S_0 K] + L[E_i S_(i-1) dotss S_0 K] + L[e_i S_(i-1) dotss S_0 K] ) \ & = (-1)^(n+1) L_(hat(K)) + z sum_(i=0)^n (-1)^i ( - L[A_n^lambda K] + L[B_n^lambda K] + L[A_i^lambda K] + L[B_i^lambda K] ) $ @@ -769,7 +773,7 @@ $ L_(K_1 union.sq K_2) = delta L_(K_1) L_(K_2) $ -per $K = K_1 union.sq K_2$ con $K_1$ sovrastante#footnote[Questo include anche il caso di componenti disgiunte in quanto sono sovrastanti tra loro a vuoto.] $K_2$. +per $K = K_1 union.sq K_2$ con $K_1$ sovrastante#footnote[Questo include anche il caso di componenti disgiunte in quanto sono sovrastanti a vuoto a vicenda.] $K_2$. sempre con $delta = (a + a^(-1)) slash z - 1$. @@ -787,7 +791,7 @@ sempre con $delta = (a + a^(-1)) slash z - 1$. kL_K (a, z) colon.eq a^w(K) $ -2. Se $K_1, K_2$ sono due diagrammi tali che $K_1 union K_2 subset.eq K$ 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 ed abbiamo che $K_1$ _sovrasta_ $K_2$, allora poniamo $ kL(K_1 union K_2) colon.eq d kL(K_1) kL(K_2) @@ -795,14 +799,14 @@ sempre con $delta = (a + a^(-1)) slash z - 1$. con $d colon.eq (a + a^(-1)) slash z - 1$. -3. Sia $K = K_1 union dotss union K_n$, se nessun $K_i$ sovrasta tutti gli altri#footnote[Questo sicuramente accade altrimenti potremmo usare il punto ii).] allora: +3. Sia $K = K_1 union dotss union K_n$ e nessun $K_i$ sovrasta tutti gli altri#footnote[Questo sicuramente accade altrimenti potremmo usare il punto ii).] 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_ il resto delle componenti. A questo punto possiamo definire $kL_K$ come + 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 #[ #set text(size: 11pt) @@ -832,9 +836,11 @@ sempre con $delta = (a + a^(-1)) slash z - 1$. == Dimostrazione buona definizione -Per prima cosa osserviamo che dato che in ogni termine della definizione induttiva per $L_K$, quando questa dipende dalla scelta di un punto base sono presenti i termini per entrambe le direzioni; quindi ci basta dimostrare induttivamente che le definizioni non dipendono dalla scelta di punto base. +// Per prima cosa osserviamo che, in quanto in ogni termine della definizione induttiva per $L_K$, quando questa dipende dalla scelta di un punto base sono presenti i termini per entrambe le direzioni; quindi ci basta dimostrare induttivamente che le definizioni non dipendono dalla scelta di punto base. -Nel corso della dimostrazione tutti gli argomenti per induzione si baseranno sul numero di incroci dei diagrammi di link. Ometteremo i casi base ovvero i controlli per diagrammi con pochi vertici che fanno partire l'induzione in quanto sono tutti facili da controllare. +Per prima cosa osserviamo che in ogni termine della definizione induttiva per $L_K$, quando questa dipende dalla scelta di un punto base, compaiono i termini per entrambe le direzioni. Dunque ci basta dimostrare induttivamente che le definizioni non dipendono dalla scelta di punto base. + +Nel corso della dimostrazione tutti gli argomenti per induzione si baseranno sul numero di incroci dei diagrammi di link. Ometteremo i casi base, ovvero i controlli per diagrammi con pochi vertici che fanno partire l'induzione in quanto sono tutte semplici verifiche. #definition( name: [L'ipotesi induttiva], @@ -859,7 +865,7 @@ Nel corso della dimostrazione tutti gli argomenti per induzione si baseranno sul ] #lemma[ - Sia $lambda = (lambda_n, dots, lambda_1, lambda_0)$ una scelta di etichette per un sottoinsieme di incroci di un diagramma di link $K$. Sia $mu = (lambda_0, lambda_n, dots, lambda_1)$ la stessa successione $lambda$ ruotata di $1$ in avanti. Allora + Sia $lambda = (lambda_n, dots, lambda_1, lambda_0)$ una scelta di etichette per un sottoinsieme di incroci di un diagramma di link $K$. Sia $mu = (lambda_0, lambda_n, dots, lambda_1)$ la stessa successione ma ruotata di $1$ in avanti. Allora $ sum_K (lambda) = sum_K (mu) @@ -867,7 +873,7 @@ Nel corso della dimostrazione tutti gli argomenti per induzione si baseranno sul ] #proof[ - Per prima cosa rinominiamo le etichette $lambda_n, dots, lambda_0$ come $n, dots, 0$ per semplificare la notazione dunque + Per prima cosa rinominiamo le etichette $lambda_n, dots, lambda_0$ come $n, dots, 0$ per semplificare la notazione, dunque $ lambda = (n, n-1, dots, 1, 0) @@ -879,20 +885,20 @@ Nel corso della dimostrazione tutti gli argomenti per induzione si baseranno sul $ sum_K (lambda) =& sum_(i=0)^n (-1) ( L[A_i^lambda K] + L[B_i^lambda K]) \ - = & + (L[E_0 K] + L[e_0 K]) + \ + = & (L[E_0 K] + L[e_0 K]) + \ & - ( L[E_1 S_0 K] + L[e_1 S_0 K]) + \ & space dots.v \ & + (-1)^n ( L[E_n S_(n-1) dotss S_0 K] + L[e_n S_(n-1) dotss S_0 K] ) $ $ sum_K (mu) = & sum_(i=0)^n (-1) (L[A_i^mu K] + L[B_i^mu K]) \ - = & + (L[E_1 K] + L[e_1 K]) + \ + = & (L[E_1 K] + L[e_1 K]) + \ & - ( L[E_2 S_1 K] + L[e_2 S_1 K]) + \ & space dots.v \ & + (-1)^n ( L[E_0 S_n dotss S_1 K] + L[e_0 S_n dotss S_1 K] ) $ - Calcoliamo ora $sum_K (lambda) - sum_K (mu)$, l'idea è di raggruppare per "righe" i termini come sono allineati qui di seguito: + Calcoliamo ora $sum_K (lambda) - sum_K (mu)$, l'idea è di fattorizzare per "righe" i termini: #{ set text(size: small-size) @@ -907,7 +913,7 @@ Nel corso della dimostrazione tutti gli argomenti per induzione si baseranno sul $ } - A questo punto per come sono allineate le somme alternate fattorizzando otteniamo quanto segue: + A questo punto per come sono allineate le somme alternate possiamo raggruppare ed otteniamo quanto segue: $ sum_K (lambda) - sum_K (mu) = + ( & L[E_0 K] + L[e_0 K]) + \ @@ -918,8 +924,6 @@ 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] ) $ - Notiamo che nella prima e nell'ultima riga abbiamo solo 2 termini che "sforano" mentre al centro abbiamo 4 termini per riga. - Permutando i termini di tutte le righe centrali possiamo riscriverli in modo da averli nella forma $L[#blank K] + L[#blank S_0 K]$: $ @@ -932,7 +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 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: $ L[#blank] + L[S_i #blank] = z (L [E_i #blank] + L[e_i #blank]) @@ -948,7 +952,7 @@ 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 le operazioni e portiamo $E_0$ e $e_0$ all'inizio delle successioni di operazioni + Riordiniamo ora le operazioni e portiamo $E_0$ e $e_0$ all'inizio delle sequenze di operazioni: #let hl(c: color.mix((white, 100%), (black, 10%)), content) = box(radius: 2pt, outset: (y: 3pt, x: 1pt), fill: c, content) @@ -961,6 +965,10 @@ 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: + $ = L[ #hl($E_0 K$)] &+ (-1)^(n+1) L[S_n dotss S_1 #hl($E_0 K$)] + \ - z lr(size: #1.5em, ( ( & L[E_1 #hl($E_0 K$)] + L[e_1 #hl($E_0 K$)])+ \ @@ -974,7 +982,7 @@ 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 $E_0 K, e_0 K$ per gli indici in comune $(n, dots, 1)$, dunque possiamo riscrivere l'espressione precedente come + 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 $ = L[ E_0 K] &+ (-1)^(n+1) L[S_n dotss S_1 E_0 K] - z sum_(E_0 K) ((n, dots, 1)) \ @@ -985,8 +993,9 @@ Nel corso della dimostrazione tutti gli argomenti per induzione si baseranno sul $ Omega_K(gamma) - &= (-1)^(abs(gamma) + 1) L_(hat(K) (gamma)) + z sum_K (gamma) \ - &= (-1)^(n+1) L_(hat(K) (gamma)) - z sum_K (gamma) + &= (-1)^(abs(gamma) + 1) L_(hat(K) (gamma)) + z sum_K (gamma) \ + &= (-1)^n L_(hat(K) (gamma)) + z sum_K (gamma) \ + => -Omega_K(gamma) &= (-1)^(n+1) L_(hat(K) (gamma)) - z sum_K (gamma) $ infine sostituendo @@ -1012,9 +1021,9 @@ Nel corso della dimostrazione tutti gli argomenti per induzione si baseranno sul ] $ - 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 $lambda(q)$ 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 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. - 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. + 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%))