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zK>$YQF&Z`~(0CYR(TPD4WB?{HNa8R;jf*dqsVI{>djn4aI^h`vI5;7FFhQ+PsQ3Ua z@En3I02nY~t{LNa$W$TyCwl`R^ZyLOB1U3R1cEY>*dfS`yf8X3+yGkQiU8&SAd8qP z0E|bUKr%2f5ds)b>H#nYF^K{gSltqtrr{Szf%hrk_a}ZasbFc$BSKJ`(6QZ`(7buz z-3NFlV&(4--Y??Ij>&i7e(wIDjbg=`kS6l~UdFHtmMvR2{vJL!#3d*sEXd8B1oVKf MkV)$5t1OKF2c`G}nE(I) diff --git a/src/main.typ b/src/main.typ index ee621b2..d3efcdc 100644 --- a/src/main.typ +++ b/src/main.typ @@ -390,27 +390,29 @@ Ora introdurremo il polinomio di Kauffman @Kauffman1990-qe, che vedremo essere u Vedremo come questi assiomi definiscono un unico invariante di isotopia regolare ben definito e come $kL_K (a, z)$ può essere generalizzato ad un invariante di isotopia ambiente $F_K (a, z)$. -*Notazione.* Per rendere alcune calcoli più leggibili, in questo capitolo utilizzeremo in modo intercambiabile le notazioni $L_K$ e $L[K]$ per indicare le valutazioni di $L$ in un certo diagramma. - == Definizione assiomatica +*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[ - Dato $K$ il diagramma di il 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 $z$, allora $kL_K in bb(Z)[a, a^(-1), z, z^(-1)]$, e verifica i seguenti assiomi: + 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 $z$. + + Allora $L_K in bb(Z)[a, a^(-1), z, z^(-1)]$, e verifica i seguenti assiomi: - 1. Se $K$, $K'$ sono equivalenti a meno di isotopia regolare, allora $kL_K = kL_K'$. + 1. Se $K$, $K'$ sono equivalenti a meno di isotopia regolare, allora $L_K = L_K'$. 2. Valgono le seguenti relazioni skein per quaterne di diagrammi identici ovunque tranne che nei punti indicati // #set enum(numbering: n => [$K_#n$)]) #set enum(numbering: "a)") - 1. $kL(#skein.over) + kL(#skein.under) = z (kL(#skein.h) + kL(#skein.v))$ + 1. $L[#skein.over] + L[#skein.under] = z (L[#skein.h] + L[#skein.v])$ - 2. $kL(#skein.unit) = 1$ + 2. $L[#skein.unit] = 1$ - 3. $kL(#skein.over-twist) = a kL(#skein.strand)$ + 3. $L[#skein.over-twist] = a L[#skein.strand]$ - 4. $kL(#skein.under-twist) = a^(-1) kL(#skein.strand)$ + 4. $L[#skein.under-twist] = a^(-1) L[#skein.strand]$ ] // 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. @@ -430,7 +432,7 @@ Come già anticipato in precedenza, vedremo che questo è un invariante di isoto ] #proof[ - Segue dalla @prop-ext-ambient-isotopy-inv. + Segue dalla @prop-ext-ambient-isotopy-inv osservando che $a$ è invertibile in $bb(Z)[a, a^(-1), z, z^(-1)]$. ] == Alcune proprietà del polinomio di Kauffman @@ -830,8 +832,6 @@ sempre con $delta = (a + a^(-1)) slash z - 1$. == Dimostrazione buona definizione -Bla bla @kauffman-rec-single-component. - 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. 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. @@ -859,7 +859,7 @@ 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 $lambda$ ma 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 $lambda$ ruotata di $1$ in avanti. Allora $ sum_K (lambda) = sum_K (mu) @@ -892,7 +892,7 @@ Nel corso della dimostrazione tutti gli argomenti per induzione si baseranno sul & + (-1)^n ( L[E_0 S_n dotss S_1 K] + L[e_0 S_n dotss S_1 K] ) $ - Calcoliamo ora $sum_K (lambda) - sum_K (mu)$, per prima cosa allineiamo i termini come segue: + Calcoliamo ora $sum_K (lambda) - sum_K (mu)$, l'idea è di raggruppare per "righe" i termini come sono allineati qui di seguito: #{ set text(size: small-size) @@ -907,7 +907,7 @@ Nel corso della dimostrazione tutti gli argomenti per induzione si baseranno sul $ } - A questo punto raggruppiamoli fattorizzando per righe: + A questo punto per come sono allineate le somme alternate fattorizzando otteniamo quanto segue: $ sum_K (lambda) - sum_K (mu) = + ( & L[E_0 K] + L[e_0 K]) + \ @@ -918,7 +918,9 @@ 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 tranne la prima e l'ultima possiamo riscriverli in modo da averli nella forma $L[#blank K] + L[#blank S_0 K]$: + 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 $L[#blank K] + L[#blank S_0 K]$: $ sum_K (lambda) - sum_K (mu) = @@ -972,7 +974,7 @@ 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 $e_0 K$, dunque possiamo riscrivere l'espressione precedente come + Infine osserviamo che i termini in $z(#blank)$ sono proprio le somme incrementali sugli scambi per $E_0 K, e_0 K$ per gli indici in comune $(n, dots, 1)$, 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)) \ @@ -997,7 +999,7 @@ Nel corso della dimostrazione tutti gli argomenti per induzione si baseranno sul E questo conclude la dimostrazione. ] -*Corollario.* Nella definizione della forma chiusa di $L_K$, il caso ii.a) non dipende dalla scelta del punto base. +*Corollario.* Nella definizione della forma chiusa di $L_K$, il caso @kauffman-rec-multi-component non dipende dalla scelta del punto base. #proof[ Rivediamo l'espressione in questione, siano $p_i$ dei punti base direzionati sulle componenti di $K$ @@ -1010,7 +1012,7 @@ 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. + 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. 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. ] @@ -1043,6 +1045,8 @@ $ #todo[work in progress] ] +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. + #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 @@ -1080,9 +1084,11 @@ $ => 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. + E questo conclude la dimostrazione delle due identità. ] +Possiamo ora mostrare l'invarianza per scelta di punto base di $Omega_K (p)$. + #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 @@ -1196,8 +1202,10 @@ $ E questo completa la dimostrazione dell'indipendenza di $Omega_K (p)$ dalla scelta di punto base. ] +A questo punto possiamo vedere che $L_K$ verifica gli assiomi della @kauffman-poly-def. + #lemma[ - Sia $i$ un incrocio di un diagramma di link $K$. Allora $L_K$ verifica le identità + Sia $i$ un incrocio di un diagramma di link $K$. Allora $L_K$ verifica le identità: 1. $L[K] + L[S_i K] = z (L[E_i K] + L[e_i K])$ @@ -1209,23 +1217,58 @@ $ 1. Consideriamo i seguenti casi: #margin-note[ricontrollare tutta questa] - - 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. A questo punto otteniamo la tesi calcolando la differenza nelle espansioni di $Omega_K (p)$ e $Omega_(S_i K) (p)$. + - 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 + + $ + 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))) + ] + $ + + 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: + + $ + => 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 di una componente con se stessa. In questo caso la tesi segue per dalla @kauffman-poly-def utilizzando per induzione il caso ii.a). + - 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. - - Se $K$ ha più di una componente e $i$ è un incrocio tra due componenti diverse allora per gli addendi che non intersecano nessuna delle due componenti possiamo procedere per induzione. Altrimenti possiamo scegliere un punto base appropriato e procedere per induzione. + - 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: + + #{ + 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))) + ] + $ + } + + 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. + + // gli addendi che non intersecano nessuna delle due componenti possiamo procedere per induzione. Altrimenti possiamo scegliere un punto base appropriato e procedere per induzione. 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. E questo conclude la dimostrazione del lemma. ] +Come ultimo risultato vediamo che $L_K$ è un invariante di isotopia regolare. + #lemma[ - Sia $K, K'$ diagrammi di link equivalenti a meno di isotopia regolare. Allora $L_K = L_(K')$, ovvero $L_K$ è un invariante di isotopia regolare. + Siano $K, K'$ diagrammi di link equivalenti a meno di isotopia regolare allora $L_K (a, z) = L_(K') (a, z)$. ] #proof[ - Procediamo sempre per induzione e mostriamo l'invarianza per mosse che non aumentano il numero di incroci. + 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]). - Mossa II: Ci sono più casi in base a se la mossa riguarda una o più componenti @@ -1243,7 +1286,7 @@ $ - Mossa III: - - Se la mossa è su fili di una sola componente allora allora possiamo scegliere il punto base sul filo che passa sopra, in questo modo avremo la situazione seguente: + - 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: #todo[disegnino] @@ -1251,7 +1294,7 @@ $ #todo[disegnino] - - Nel caso in cui la mossa riguardi fili di più componenti possiamo utilizzare l'identità della mossa precedente $L[K] = L[S_2 S_1 K]$ per semplificare la situazione e 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. Questo completa la dimostrazione. ] \ No newline at end of file