From ff05f7996a381a7a576f2e95e9bfa23f682f5137 Mon Sep 17 00:00:00 2001 From: Antonio De Lucreziis Date: Wed, 9 Jul 2025 01:46:25 +0200 Subject: [PATCH] more work --- out/tesi-triennale.pdf | Bin 1759396 -> 2171555 bytes src/assets/modified-curl-add-after.png | Bin 0 -> 173090 bytes src/assets/modified-curl-add-before.png | Bin 0 -> 67420 bytes src/assets/modified-r2-after.png | Bin 0 -> 183351 bytes src/assets/modified-r2-before.png | Bin 0 -> 102575 bytes src/main.typ | 123 +++++++++++++++++------- 6 files changed, 86 insertions(+), 37 deletions(-) create mode 100644 src/assets/modified-curl-add-after.png create mode 100644 src/assets/modified-curl-add-before.png create mode 100644 src/assets/modified-r2-after.png create mode 100644 src/assets/modified-r2-before.png diff --git a/out/tesi-triennale.pdf b/out/tesi-triennale.pdf index 3f67d8c8f8e0644fb2443f426e3a141952fee43f..0c93a05f4aa31bcd340716eca6bd262aa8c3fc26 100644 GIT binary patch delta 474672 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In particolare questo ci permette anche di "fattorizzare" tutti i riccioli di una componente in modo che siano tutti vicini. -Un'altra osservazione sul comportamento dell'isotopia regolare rispetto alle mosse di tipo I è la seguente. Quando abbiamo una coppia di riccioli con segni opposti abbiamo la seguente proprietà di cancellazione detto _trucco di Whitney_. +#observation[ + Un'altra osservazione sul comportamento dell'isotopia regolare rispetto alle mosse di tipo I è la seguente. Quando abbiamo una coppia di riccioli con segni opposti abbiamo la seguente proprietà di cancellazione detto _trucco di Whitney_. +] #figure(image("assets/whitney-trick.png", width: 16cm)) @@ -688,35 +690,46 @@ Enunciamo ora un lemma che utilizzeremo più volte in seguito. ] #proof[ - Per il @std-unknot-trivial-knot, $K$ è un nodo banale quindi possiamo applicare il teorema di Reidemeister ed ottenere una successione di mosse e diagrammi tale che + Per il @std-unknot-trivial-knot, un nodo banale standard è equivalente ad un nodo banale a meno di isotopia ambiente. Quindi possiamo applicare il teorema di Reidemeister ed ottenere una successione di mosse e diagrammi tale che $ K = D_0 stretch(arrow)^(m_1) dots.c stretch(arrow)^(m_n) D_n = #skein.unit-medium $ - con $m_i$ mosse di tipo I, II, III ovvero che porta il diagramma di $K$ al diagramma banale. Ora l'idea è di modificare questa successione di mosse in modo da ottenere una sequenza composta solo da mosse II e III e rimanere con un diagramma composto solo da riccioli. + con $m_i$ mosse di tipo I, II, III. Ora l'idea modificare questa successione di mosse in modo da ottenere una nuova successione composta solo da mosse II e III che termina con un diagramma composto solo da riccioli. + + Applichiamo induttivamente le seguenti modifiche ad ogni mossa $m_i$ nella sequenza, tenendo traccia di cosa succede man mano agli incroci: - Applichiamo induttivamente le seguenti modifiche per ogni mossa $m_i$ nella sequenza: + - Se abbiamo una mossa di tipo I che rimuove un ricciolo, allora scartiamo la mossa in modo da conservare il ricciolo. In particolare in tutte le mosse successive non utilizzeranno questo incrocio in quanto nella successione originale veniva cancellato quindi questo sarà uno di quei riccioli che rimarrà fino al diagramma finale. - - Se la mossa è di tipo I e rimuove un ricciolo allora la scartiamo e conserviamo il ricciolo. In particolare in tutte le mosse successione non utilizzeranno l'incrocio formato da questo ricciolo in quanto nella successione originale veniva eliminato. + - Se abbiamo una mossa di tipo I che aggiunge un ricciolo, allora possiamo introdurlo sfruttando l'@obs-whitney-trick ovvero il trucco di Whitney. Possiamo creare due riccioli utilizzando solo mosse II e III in modo tale che uno dei due incroci sia quello introdotto dal ricciolo della sequenza originale mentre l'altro sarà in più ma possiamo ignorarlo come nel caso precedente. - - Se la mossa è di tipo I e aggiunge un ricciolo allora possiamo modificarla utilizzando il _trucco di Whitney_ in modo da crearlo utilizzando solo mosse II e III come mostrato in figura. L'unica accortezza è di applicare il giusto ricciolo per compensare quello di partenza come mostrato in figura + #figure(image("assets/modified-curl-add-before.png", width: 6.5cm)) + #figure(image("assets/modified-curl-add-after.png", width: 13cm)) - #todo[disegnino] + Abbiamo aggiunto così due incroci uno dei quali corrisponde ad uno della sequenza originale mentre l'altro è uno di quei riccioli che rimarranno nel diagramma finale. - - Se la mossa è di tipo II o III allora utilizziamo l'@curls-pass-through per spostare tutti i riccioli che si trovano nella regione in cui vogliamo applicare la mossa in questione al di fuori di essa. + - Se abbiamo una mossa di tipo II o III, allora questa mossa interesserà una certa regione del diagramma. Per prima cosa utilizziamo l'@obs-curls-pass-through per spostare tutti quei riccioli che si trovano nella regione in questione al di fuori di essa e poi applichiamo la mossa originale. + + #figure(image("assets/modified-r2-before.png", width: 6.5cm)) + #figure( + image("assets/modified-r2-after.png", width: 8.5cm), + caption: [Esempio per la mossa II che rimuove un incrocio], + ) - #todo[disegnino] + In questo caso abbiamo solo spostato dei riccioli ed applicato la stessa mossa della sequenza originale quindi non abbiamo introdotto incroci in più rispetto alla sequenza originale. - Otteniamo così una nuova successione di diagrammi $D_i^*$ e di mosse solo di tipo II o III + Otteniamo così una nuova successione di diagrammi $D'_i$ e di mosse $m_i^*$ solo di tipo II o III: $ - K = D_0^* stretch(arrow)^(m_1^*) dots.c stretch(arrow)^(m_n^*) D_n^* + K = D'_0 stretch(arrow)^(m_1^*) dots.c stretch(arrow)^(m_n^*) D'_n $ - Per conclude osserviamo che c'è una corrispondenza tra gli incroci di $D_i$ e $D_i^*$ che ci induce anche una corrispondenza tra $D_(i+1)$ e $D_(i+1)^*$. In particolare nella successione originale man mano rimuoviamo tutti gli incroci. Tutti gli incroci si cancellano anche nella nuova successione tranne per quelli che abbiamo aggiunto modificando le mosse di tipo I, ma tutti questi sono dei riccioli quindi $D_n^*$ sarà composto solo da riccioli. + Vorremo ora vedere che il diagramma $D'_n$ è composto solo da riccioli. Questo segue dal fatto che nella successione originale tutti gli incroci si cancellavano, ciò accade anche nella nuova successione tranne per gli incroci che abbiamo aggiunto modificando le mosse di tipo I, ma tutti questi sono dei riccioli quindi $D'_n$ sarà composto solo da riccioli. ] +// che nella successione originale man mano rimuoviamo tutti gli incroci. Tutti gli incroci si cancellano anche nella nuova successione tranne per quelli che abbiamo aggiunto modificando le mosse di tipo I, ma tutti questi sono dei riccioli quindi $D_n^*$ sarà composto solo 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. Per prima cosa analizziamo quello che sarà il caso base della nostra definizione induttiva, ovvero quando abbiamo un nodo banale standard. 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 $L_hat(K) (a, z) = a^w(K)$. @@ -1089,10 +1102,18 @@ La dimostrazione procede sempre mostrando l'indipendenza dalla scelta di punto b 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 discendente). ] +#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] +] + // #grid( // columns: (1fr, auto), // gutter: 1em, @@ -1138,11 +1159,26 @@ La dimostrazione procede sempre mostrando l'indipendenza dalla scelta di punto b ] #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. + 1. In $hat(K)(cal(U), p)$, l'incrocio $i$ è il primo visitato quindi sarà sicuramente un sopra-incrocio. Al contrario, in $hat(K)(cal(U), q)$ l'incrocio $i$ è l'ultimo visitato quindi è certamente la seconda occorrenza di $i$ e sarà un sotto-incrocio. Tutti gli altri incroci compaiono nello stesso ordine quindi restano invariati. Dunque $S_i hat(K) = hat(K)'$. + + #{ + set align(center) + set text(size: small-size) - 2. Per questo secondo punto usiamo che $L_hat(K) = a^w(hat(K))$, assumiamo ora che tra $E_i hat(K)$ sia il caso con due componenti di due nodi banali standard e $e_i hat(K)$ quello con una sola componente di diagramma di nodo banale. + grid( + columns: 2, + row-gutter: 0.5em, + column-gutter: 2em, + image("assets/derived/atlas-crossing-slides-0.png", width: 2.25cm), + image("assets/derived/atlas-crossing-slides-3.png", width: 2.25cm), + $hat(K)(cal(U), p)$, + $hat(K)(cal(U), q)$, + ) + } - Supponiamo che $w(e_i hat(K)) = w$, allora in quanto abbiamo solo rimosso un incrocio necessariamente + 2. Per questo secondo punto usiamo che $L_hat(K) = a^w(hat(K))$, assumiamo ora che $E_i hat(K)$ sia il caso con due componenti di due nodi banali standard e $e_i hat(K)$ quello con una sola componente di diagramma di nodo banale. + + Supponiamo che $w(e_i hat(K)) = w$, allora poiché abbiamo rimosso un incrocio vale necessariamente $ {w(hat(K)), w(S_i hat(K))} = {w-1, w+1} @@ -1150,7 +1186,9 @@ La dimostrazione procede sempre mostrando l'indipendenza dalla scelta di punto b 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 $w_1 + w_2 = w$. + Nell'altro caso avremo due componenti, ovvero $E_i hat(K) = K_1 union.sq K_2$, per come sono definite le mosse $E_i$ e $e_i$ abbiamo anche che $e_i hat(K) = K_1 hash K_2$. + + Siano ora $w_1$ e $w_2$ i writhe rispettivamente di $K_1$ e $K_2$. 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 da cui $w_1 + w_2 = w$. Ricapitoliamo ora tutti i termini: $ L[hat(K)] &= a^(w+1) \ @@ -1159,7 +1197,7 @@ La dimostrazione procede sempre mostrando l'indipendenza dalla scelta di punto b L[E_i hat(K)] &= delta a^(w_1) a^(w_2) = delta a^(w_1 + w_2) = delta a^w = (a^(w+1) + a^(w-1)) slash z - a^w $ - che dunque verifica l'identità: + che verifica l'identità infatti: $ a^(w+1) + a^(w-1) &= cancel(z) ((a^(w+1) + a^(w-1))/cancel(z) - cancel(a^w) + cancel(a^w)) \ @@ -1182,13 +1220,13 @@ Possiamo ora mostrare l'invarianza per scelta di punto base di $Omega_K (p)$. ] #proof[ - Possiamo assumere che la sequenza di scambi determinata da $p$ sia etichetta $lambda = (n, dots, 0)$, in questo modo + Possiamo assumere che la sequenza di scambi determinata da $p$ sia etichettata come $lambda = (n, dots, 0)$, in questo modo $ Omega_K (p) = (-1)^(n+1) L_(hat(K)(p)) + z sum_K (lambda(p)) $ - Ricordiamo come era definito $L_K$ nel caso di una sola componente @kauffman-rec-single-component e notiamo quanto segue + Ricordiamo come era definito $L_K$ nel caso @kauffman-rec-single-component ovvero di una sola componente e notiamo quanto segue $ L_K (a, z) &colon.eq 1 / 2 [ @@ -1200,19 +1238,26 @@ Possiamo ora mostrare l'invarianza per scelta di punto base di $Omega_K (p)$. Dunque ci basta mostrare che $Omega_K (p)$ non dipenda dalla scelta di punto base. - Come in precedenza possiamo mostrarlo facendo scorrere $p$ nell'arco successivo al primo incrocio dopo $p$ (questo ci induce l'invarianza per permutazioni cicliche e ci permette di concludere). + Come in precedenza possiamo mostrarlo facendo scorrere $p$ nell'arco successivo al primo incrocio dopo $p$, questo ci induce l'invarianza per permutazioni cicliche e ci permette di concludere. - Sia $i$ l'etichetta del primo incrocio dopo $p$. In $hat(K)(p)$ questo incrocio sarà sicuramente un sopra-incrocio in quanto $i$ è visitato per la prima volta essendo il primo incrocio dopo $p$. Invece in $K(p)$ può essere sia un sopra-incrocio che un sotto-incrocio, abbiamo quindi due casi: + Sia $i$ l'etichetta del primo incrocio dopo $p$. In $hat(K)(p)$ questo incrocio sarà sicuramente un sopra-incrocio in quanto $i$ è il primo incrocio visitato dopo $p$. Invece in $K(p)$ può essere sia un sopra-incrocio che un sotto-incrocio, abbiamo quindi due casi: #{ set align(center) + set text(size: small-size) grid( - columns: 2, + columns: 3, row-gutter: 0.5em, column-gutter: 2em, - image("assets/derived/atlas-crossing-slides-0.png", width: 2.5cm), - image("assets/derived/atlas-crossing-slides-1.png", width: 2.5cm), + align: center + horizon, + image("assets/derived/atlas-crossing-slides-1.png", width: 2.25cm), + [ + #set text(size: footnote-size) + oppure + ], + image("assets/derived/atlas-crossing-slides-0.png", width: 2.25cm), $K(p)$, + [], $K(p)$, ) } @@ -1221,12 +1266,13 @@ Possiamo ora mostrare l'invarianza per scelta di punto base di $Omega_K (p)$. #{ set align(center) + set text(size: small-size) grid( columns: 2, row-gutter: 0.5em, column-gutter: 2em, - image("assets/derived/atlas-crossing-slides-1.png", width: 2.5cm), - image("assets/derived/atlas-crossing-slides-0.png", width: 2.5cm), + image("assets/derived/atlas-crossing-slides-1.png", width: 2.25cm), + image("assets/derived/atlas-crossing-slides-0.png", width: 2.25cm), $K(p)$, $hat(K)(p)$, ) @@ -1236,18 +1282,19 @@ Possiamo ora mostrare l'invarianza per scelta di punto base di $Omega_K (p)$. #{ set align(center) + set text(size: small-size) grid( columns: 2, row-gutter: 0.5em, column-gutter: 2em, - image("assets/derived/atlas-crossing-slides-3.png", width: 2.5cm), - image("assets/derived/atlas-crossing-slides-3.png", width: 2.5cm), + image("assets/derived/atlas-crossing-slides-3.png", width: 2.25cm), + image("assets/derived/atlas-crossing-slides-3.png", width: 2.25cm), $K(q)$, $hat(K)(q)$, ) } - come già detto prima se l'ultimo incrocio è un sotto-incrocio allora è già in forma di nodo banale standard. Segue che $i$ non appartiene alla sequenza di scambi in questo caso ed avremo $(n-1, dots, 0)$ e $hat(K)(q) = S_(n-1) dotss S_0 K$. Vorremo vedere che $Omega_K (p) = Omega_K (q)$, ovvero: + come già detto prima l'ultimo incrocio è un sotto-incrocio dunque $K(q)$ è già in forma di nodo banale standard. Segue che $i$ non appartiene alla sequenza di scambi e in questo caso avremo $(n-1, dots, 0)$ e $hat(K)(q) = S_(n-1) dotss S_0 K$. Vorremo vedere che $Omega_K (p) = Omega_K (q)$, ovvero: $ Omega_K (p) &= (-1)^(n+1) L[hat(K)(p)] + z sum_K (lambda(p)) \ @@ -1260,6 +1307,8 @@ Possiamo ora mostrare l'invarianza per scelta di punto base di $Omega_K (p)$. Omega_K (p) - Omega_K (q) =& (-1)^(n+1) L[hat(K)(p)] + z sum_K (lambda(p)) \ &+ underbrace((-1)^(n+1), = -(-1)^n) L[hat(K)(q)] - z sum_K (lambda(q)) \ + $ + $ =& (-1)^(n+1) (L[hat(K)(p)] + L[hat(K)(q)]) \ &+ z (-1)^n (L[A_n^lambda K] + L[B_n^lambda K]) $ @@ -1280,7 +1329,7 @@ Possiamo ora mostrare l'invarianza per scelta di punto base di $Omega_K (p)$. L[hat(K)(p)] + L[hat(K)(q)] = z (L[E_n hat(K)(p)] + L[e_n hat(K)(p)]) $ - Per concludere questo caso basta sostituire in queste ultime identità + Per concludere basta sostituire queste ultime identità $ Omega_K (p) - Omega_K (q) =& (-1)^(n+1) (L[hat(K)(p)] + L[hat(K)(q)]) \ @@ -1300,14 +1349,14 @@ Possiamo ora mostrare l'invarianza per scelta di punto base di $Omega_K (p)$. columns: 2, row-gutter: 0.5em, column-gutter: 2em, - image("assets/derived/atlas-crossing-slides-0.png", width: 2.5cm), - image("assets/derived/atlas-crossing-slides-0.png", width: 2.5cm), + image("assets/derived/atlas-crossing-slides-0.png", width: 2.25cm), + image("assets/derived/atlas-crossing-slides-0.png", width: 2.25cm), $K(p)$, $hat(K)(p)$, ) } - In questo caso, $i$ non fa parte della sequenza di scambi per $K(q)$. Però dopo aver spostato il punto base avremo la seguente situazione + In questo caso, $i$ non fa parte della sequenza di scambi per $K(p)$. Però dopo aver spostato il punto base avremo la seguente situazione #{ set align(center) @@ -1315,14 +1364,14 @@ Possiamo ora mostrare l'invarianza per scelta di punto base di $Omega_K (p)$. columns: 2, row-gutter: 0.5em, column-gutter: 2em, - image("assets/derived/atlas-crossing-slides-2.png", width: 2.5cm), - image("assets/derived/atlas-crossing-slides-3.png", width: 2.5cm), + image("assets/derived/atlas-crossing-slides-2.png", width: 2.25cm), + image("assets/derived/atlas-crossing-slides-3.png", width: 2.25cm), $K(q)$, $hat(K)(q)$, ) } - Questa volta $i$ fa parte della sequenza di scambi per $K(q)$ e quando compare come ultimo incrocio è un sotto-incrocio. Avremo quindi le seguenti sequenze di scambi + Questa volta $i$ fa parte della sequenza di scambi per $K(q)$ in quanto compare come ultimo incrocio dove sarà un sotto-incrocio. Stavolta la situazione è più complessa poiché $i$ compare come sopra-incrocio in mezzo alla sequenza di scambi e non necessariamente all'inizio o alla fine. Avremo quindi le seguenti sequenze per $p$ e per $q$: $ lambda(q) &= (n, n-1, dots, i, dots, 1, 0) \