From 40a466859c683d46bafa2d044e92aa67ca4221a1 Mon Sep 17 00:00:00 2001 From: Antonio De Lucreziis Date: Sun, 13 Jul 2025 23:30:21 +0200 Subject: [PATCH] versione finale --- out/tesi-triennale.pdf | Bin 3119293 -> 3118411 bytes src/main.typ | 99 ++++++++++++++++++++++------------------- 2 files changed, 54 insertions(+), 45 deletions(-) diff --git a/out/tesi-triennale.pdf b/out/tesi-triennale.pdf index fdbf79d30974da2a7b3100486a82ab4f184e5a99..9899013daf88e82a644f0fc8d421310488302eed 100644 GIT binary patch delta 138203 zcmZs>1yEdFu&y24HMqOGOK=PB?oM!m1>M2jEjR?X;O_1o+}&M*^AF!S|E;=JcdDLQ zd)Dsmx4YMxnLW~8-2paQ+5v`6sU$AR$jZbGPx%vcgNNtfP9Wu@NN`;F94D;J|{*o4JNxqxAUPB2`d zNHFBXOX%X|#bW<|NpmLX^KyNtfy_d|I6jnW_?f`LI1|8xF>1vG+QCz*y_qp-xA&w0O$ND z6FYtmy0+N#0s`P_@A5>8&w92+PO8yg7+@5dZF2o4f1u8)!PKmOBV1H}Qs!~QYP4Sd{@uycGE zfCNEsvww^q0eDF`xIo72paem1@_-!JK>{Fn*g<0KAaM}fpk-zK2nK?S8|2FlN*V;` zeg_ur81pIIU33Gx(+5ZO)PS6|&C@=>r3CBmZ93U|e z|1%E~<6tM@235-X(J>Hg9|9jmfcQ_010)85>!Z6IAO74VoZKICA9f%uP*+(wKw1Fk 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zX0Z_sNc4XOV$`8~7h>tD|07Jt5sV59P zG~9szk4gj=2hibysG~*{n=l)NV{Nfv6HqQVBL@Bjjx2zBAWAm~G` zv-3a1BDg9Tv=srT{c+j?jq=|d|F10yo7)SAGa!h^v4EgM5G&!ZAHYQZIx@#VrT;@92&Hj6InWTGIjXX*U=XFn}lFYuEBQUv?^ z#8n3euzQc4=HEX8+FLm=69^&Un3cy6F8Key&nh9o{quK47gtkfS5Ie4D^e^*pg>7E JIONo?{2zqWA=Ur@ diff --git a/src/main.typ b/src/main.typ index 4e1d7ba..a41033a 100644 --- a/src/main.typ +++ b/src/main.typ @@ -282,7 +282,9 @@ C'è un'osservazione importante da fare quando parliamo di isotopia regolare. Qu // 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. -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 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. @@ -291,9 +293,9 @@ Questo sarà importante più avanti quando vedremo invarianti come il polinomio #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. + 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. + 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. #h(1fr) #figure( image("assets/standard-unlink-construction.png", width: 12cm), @@ -479,7 +481,7 @@ Vedremo come questi assiomi definiscono un unico invariante di isotopia regolare 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 nella parte evidenziata + 2. Valgono le seguenti relazioni skein per quaterne di diagrammi identici ovunque tranne nella parte evidenziata: // #set enum(numbering: n => [$K_#n$)]) #set enum(numbering: "a)") @@ -500,7 +502,7 @@ Al momento non sappiamo se il polinomio $L_K$ sia ben definito, ovvero se esiste In questo lavoro vedremo che $L_K$ esiste ed è unico e che è un 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 di link orientati $K$ come + Definiamo $F_K in bb(Z)[a, a^(-1), z, z^(-1)]$ per diagrammi di link orientati $K$ come $ F_K colon.eq a^(-w(K)) kL_K $ @@ -579,7 +581,7 @@ $ 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 +Possiamo anche trovare il valore per il link di Hopf come segue: // #figure(image("assets/implicit-calc-2.png")) @@ -754,19 +756,19 @@ Sia $lambda = (lambda_n, dots, lambda_0)$ una sequenza di etichette di incroci d Dati $K_1, K_2$ diagrammi di link, diciamo che $K_1$ *sovrasta* $K_2$ se ogni incrocio in $K_1 inter K_2$ è un sopra-incrocio. ] -#diff-add[ - Prima di procedere consideriamo anche una generalizzazione dei diagrammi in forma discendente che utilizzando le cosiddette funzioni di slacciamento. Queste possono essere pensate come funzioni che ci danno una "altezza" del diagramma in ogni punto. +// #diff-add[ +Prima di procedere consideriamo anche una generalizzazione dei diagrammi in forma discendente che utilizza le cosiddette funzioni di slacciamento. Queste possono essere pensate come funzioni che ci danno una "altezza" del diagramma in ogni punto. - #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: +#definition[ + Dato $D$ il diagramma di un link $L$, una *funzione di slacciamento* per $D$ è una mappa $h$ da $D$ a valori in $RR$, che assume due valori in corrispondenza degli incroci e che corrisponde a una funzione continua $h : L -> RR$. Tale mappa ha le seguenti proprietà: - 1. Se una componente $C_i$ sovrasta una componente $C_j$ allora $forall x_i in C_i$ e $forall x_j in C_j$, $h(C_i) > h(C_j)$. + 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. 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 $C_i$. + 2. Su ogni componente $C_i$ esistono dei punti $b_i, t_i in C_i$ tali che la funzione $h$ è strettamente monotona crescente da $b_i$ a $t_i$ in entrambe le direzioni attorno a $C_i$. - 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 $h_+ > h_-$. - ] + 3. In corrispondenza di un incrocio, $h$ assume due valori $h_+, h_-$ rispettivamente nel caso in cui passi da sopra o da sotto e tali che $h_+ > h_-$. ] +// ] // #align(center, line(stroke: 0.5pt, length: 50%)) @@ -933,7 +935,7 @@ sempre con $delta = (a + a^(-1)) slash z - 1$. 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. +Nel corso della dimostrazione tutti gli argomenti per induzione si baseranno sul numero di incroci. Ometteremo i casi base, ovvero i controlli per diagrammi con pochi incroci che fanno partire l'induzione in quanto sono tutte semplici verifiche. #definition( numbered: true, @@ -957,7 +959,8 @@ Nel corso della dimostrazione tutti gli argomenti per induzione si baseranno sul 3. #marker([c)]) $L_K$ è invariante per mosse di tipo II e III che non aumentano il numero di incroci. - 4. #diff-add[#marker([d)]) Se $K$ ammette una _funzione di slacciamento_ allora $L_K = a^w(K)$] + // 4. #diff-add[#marker([d)]) Se $K$ ammette una _funzione di slacciamento_ allora $L_K = a^w(K)$] + 4. #marker([d)]) Se $K$ ammette una _funzione di slacciamento_ allora $L_K = a^w(K)$ ] #lemma[ @@ -1155,47 +1158,49 @@ La dimostrazione procede sempre mostrando l'indipendenza dalla scelta di punto b // Questo segue semplicemente sommando le espressioni per $L[K]$ e $L[S_i K]$. -#diff-add[ - Enunciamo ora il seguente enunciato nel caso di aver definito $L_K$ rispetto ad una sola orientazione ovvero, dato $p$ punto base e $i$ un indice nella sequenza di scambi indotta da $p$: +// #diff-add[ +Enunciamo ora il seguente enunciato nel caso di aver definito $L_K$ rispetto ad una sola orientazione ovvero, dato $p$ punto base e $i$ un indice 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_i K] + L[e_i K]) - L[S_i K] - $ +$ + L_K (a, z) & colon.eq + (-1)^(abs(lambda(p))+1) L[hat(K)(lambda(p))] + z sum_K (lambda(p)) \ + & = z (L[E_i K] + L[e_i K]) - L[S_i K] +$ - l'ultima uguaglianza segue considerando $L[K]$ e $L[S_i K]$ e sottraendo membro a membro. +// l'ultima uguaglianza segue considerando $L[K]$ e $L[S_i K]$ e sottraendo membro a membro. - #fact[ - Se $K$ è un diagramma di link e ammette una funzione di slacciamento allora $L_K = a^w(K)$. - ] +#proposition(numbered: false)[ + Se $K$ è un diagramma di link e ammette una funzione di slacciamento allora $L_K = a^w(K)$. +] - Per una dimostrazione di questo fatto si rimanda a @lickorish1997introduction[pag. 175-176]. +Per una dimostrazione di questo fatto si rimanda a @lickorish1997introduction[pag. 175-176]. - // #proof[ - // 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. +// #proof[ +// 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. +// 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 $b$, ci sono due casi in base a come appare partendo da $b$: +// Consideriamo ora una singola componente, sia $i$ il primo incrocio dopo il punto a valle $b$, ci sono due casi in base a come appare partendo da $b$: - // - Se è un sotto-incrocio, $h$ può essere modificata in modo da essere decrescente da $b$ fino a subito dopo $i$ in modo che sia ancora una funzione di slacciamento. +// - Se è un sotto-incrocio, $h$ può essere modificata in modo da essere decrescente da $b$ fino a subito dopo $i$ in modo che sia ancora una funzione di slacciamento. - // Dunque $L_K = a^w(K)$ per sotto-induzione in quanto $K$ con la nuova funzione di slacciamento ha il punto $b$ di un incrocio più vicino al punto di base. +// Dunque $L_K = a^w(K)$ per sotto-induzione in quanto $K$ con la nuova funzione di slacciamento ha il punto $b$ di un incrocio più vicino al punto di base. - // - Se è un sopra-incrocio deve essere necessariamente un punto in cui una parte della componente in cui andiamo dal punto base verso $b$. Segue dalle proprietà di monotonicità di $h$. +// - Se è un sopra-incrocio deve essere necessariamente un punto in cui una parte della componente in cui andiamo dal punto base verso $b$. Segue dalle proprietà di monotonicità di $h$. - // In questo caso calcoliamo $L_K$ usando la proprietà $(*)$ applicata all'incrocio $i$. +// In questo caso calcoliamo $L_K$ usando la proprietà $(*)$ applicata all'incrocio $i$. - // - Scambiare l'incrocio dà un diagramma $K'$ in cui possiamo spostare il punto $b$ più vicino al punto base e dunque per induzione $L_K' = a^w(K')$. +// - Scambiare l'incrocio dà un diagramma $K'$ in cui possiamo spostare il punto $b$ più vicino al punto base e dunque per induzione $L_K' = a^w(K')$. - // - I diagrammi con i due tipi di splice hanno $n-1$ incroci ed ammettono delle funzioni di slacciamento, dunque anche loro sono noti per induzione - // ] +// - I diagrammi con i due tipi di splice hanno $n-1$ incroci ed ammettono delle funzioni di slacciamento, dunque anche loro sono noti per induzione +// ] - #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. - ] +#proposition( + numbered: false, +)[ + L'enunciato precedente si generalizza anche con la nostra definizione di $L_K$ che prende in considerazione entrambe le direzioni. Si può dimostrare procedendo in modo analogo ma serve tenere conto di come si presentano gli incroci in entrambe le direzioni. ] +// ] #lemma[ Consideriamo i due modi di fare uno splice di un nodo banale standard al primo incrocio subito dopo un punto base direzionato. Allora valgono le seguenti proprietà: @@ -1225,12 +1230,16 @@ La dimostrazione procede sempre mostrando l'indipendenza dalla scelta di punto b [], $hat(K)$, $E_i hat(K)$, - $E_i hat(K)$, + $e_i hat(K)$, [], )), ) - 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]. + Osserviamo che passeremo prima dalla componente $K_2^*$ in verso opposto, questa parte sarà quindi in forma ascendente. Al secondo incontro dello splice dell'incrocio $i$, torneremo sulla componente $K_1$ che percorreremo nel verso originale che sarà quindi 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 di massimo $t$ e come punto di minimo $b$ come mostrato nella figura precedente nel caso di $e_i hat(K)$. I colori indicano, in riferimento alle orientazioni, quelle parti che sono rispettivamente monotone #box(radius: 3pt, outset: (y: 3pt), inset: (x: 2pt), fill: green.mix(white))[crescenti] o #box(radius: 3pt, outset: (y: 3pt), inset: (x: 2pt), fill: red.mix(white))[decrescenti]. ] // #diff-del[ @@ -1277,7 +1286,7 @@ La dimostrazione procede sempre mostrando l'indipendenza dalla scelta di punto b // *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 +*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 ed ha un incrocio in meno, 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))