From 841ea6cdc23ad5b93c7b55f1e0afc28268091842 Mon Sep 17 00:00:00 2001 From: Antonio De Lucreziis Date: Wed, 9 Jul 2025 21:01:05 +0200 Subject: [PATCH] almost finished --- out/tesi-triennale.pdf | Bin 2272292 -> 2834805 bytes src/assets/atlas-move-2-K.png | Bin 0 -> 163621 bytes src/assets/atlas-move-2-S1K.png | Bin 0 -> 159794 bytes src/assets/atlas-move-3-equivalences.png | Bin 0 -> 189122 bytes src/assets/atlas-move-3.png | Bin 0 -> 183828 bytes src/assets/derived/atlas-move-2-K-0.png | Bin 0 -> 19852 bytes src/assets/derived/atlas-move-2-K-1.png | Bin 0 -> 19735 bytes src/assets/derived/atlas-move-2-K-2.png | Bin 0 -> 18945 bytes src/assets/derived/atlas-move-2-K-3.png | Bin 0 -> 20501 bytes src/assets/derived/atlas-move-2-S1K-0.png | Bin 0 -> 19854 bytes src/assets/derived/atlas-move-2-S1K-1.png | Bin 0 -> 20388 bytes src/assets/derived/atlas-move-2-S1K-2.png | Bin 0 -> 18919 bytes src/assets/derived/atlas-move-2-S1K-3.png | Bin 0 -> 19426 bytes 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Tagliando questi due archi e riunendo insieme le estremità dello stesso sotto-diagramma otteniamo un nuovo diagramma split della forma $tilde(K) = K_1 union.sq K_2$, in questo caso chiamiamo la somma connessa $K = K_1 hash K_2$. ] -#todo[aggiungere figura] +#figure( + image("assets/split-and-connected-sum.png", width: 10cm), + caption: [Esempio di diagrammi _split_ ed in _somma connessa_], +) In particolare abbiamo le seguenti identità per i polinomi $L_K$ e $F_K$ in relazione ai diagrammi _split_ e in _somma connessa_. La dimostrazione è omessa ma si fa procedendo per induzione sul numero di incroci. @@ -700,14 +703,14 @@ Enunciamo ora un lemma che utilizzeremo più volte in seguito. Applichiamo induttivamente le seguenti modifiche ad ogni mossa $m_i$ nella sequenza, tenendo traccia di cosa succede man mano agli incroci: - - 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 della sequenza originale non utilizzeranno questo incrocio in quanto questa mossa lo cancellava quindi questo sarà uno di quei riccioli che rimarrà nel nuovo diagramma finale. + - 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 della sequenza originale, sicuramente non compare questo incrocio poiché questa mossa lo cancellava. Quindi questo sarà uno di quei riccioli che rimarrà nel nuovo diagramma finale. - 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 sia orientato allo stesso modo di quello introdotto dal ricciolo della sequenza originale, l'altro sarà in più ma possiamo ignorarlo come nel caso precedente. #figure(image("assets/modified-curl-add-before.png", width: 6.5cm)) #figure(image("assets/modified-curl-add-after.png", width: 13cm)) - 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. + Abbiamo aggiunto così due incroci uno dei quali corrisponde ad uno della vecchia sequenza che verrà cancellato dalle mosse originali mentre l'altro è uno di quei riccioli che rimarranno nel diagramma finale. - 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. @@ -728,6 +731,10 @@ Enunciamo ora un lemma che utilizzeremo più volte in seguito. 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. ] +*Corollario.* Un'altro modo di vedere il lemma precedente è che il diagramma di un nodo in forma discendente è equivalente a meno di isotopia regolare ad uno composto solo da riccioli. + +#align(center, line(stroke: 0.5pt, length: 50%)) + // 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. @@ -910,7 +917,7 @@ Nel corso della dimostrazione tutti gli argomenti per induzione si baseranno sul - $L[#skein.under-twist-medium] = a^(-1) L [#skein.strand-medium]$ - 3. $L_K$ è invariante per mosse di tipo II e III che non aumentano il numero di incroci. + 3. #marker([c)]) $L_K$ è invariante per mosse di tipo II e III che non aumentano il numero di incroci. ] #lemma[ @@ -1099,25 +1106,25 @@ $ La dimostrazione procede sempre mostrando l'indipendenza dalla scelta di punto base. Vediamo prima alcune proprietà dei nodi banali standard: #lemma[ - 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). + 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: + + 1. Nei due casi, in uno otteniamo _due nodi banali in forma standard_, nell'altro _un solo nodo banale_ (non necessariamente in forma discendente). + + 2. La proprietà del lemma @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. ] #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 dopo la seconda occorrenza di $i$). - - #figure(image("assets/splice-circuits.png", width: 4cm)) + 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 $i$). 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 che la prima volta che si passa su un incrocio questo è un sopra-incrocio. - Se assumiamo che il link del caso precedente sia della forma $E_i K = K_1 union K_2$ con $K_1$, $K_2$ i due nodi banali standard allora l'altro caso sarà $e_i K = K_1 hash K_2$. 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. Dunque se $K_1$ e $K_2$ sono equivalenti a diagrammi formati solo da riccioli lo sarà anche $K_1 hash K_2$ sfruttando il fatto che uno dei due diagrammi sovrasta l'altro e possiamo spostarlo utilizzando solo mosse di tipo II o III#margin-note[Prima dire che possiamo ricondurci alla situazione $K_1 union.sq K_2$ e poi incollare]. -] + Se assumiamo che il link 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 da $hat(K)_2$ percorrendolo in direzione opposta. -#lemma[ - La proprietà del lemma @std-unknot-to-curls si estende al diagramma $e_i hat(K)$. -] + #figure(image("assets/splice-circuits.png", width: 4cm)) -#proof[ - #todo[work in progress] + 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 senza problemi. 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( @@ -1154,6 +1161,8 @@ La dimostrazione procede sempre mostrando l'indipendenza dalla scelta di punto b // #todo[work in progress] // ] +*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 $L_(e_i hat(K)) = a^w(e_i hat(K))$. + // Utilizzando questi ultimi risultati identità per nodi banali standard che ci mostrano cosa succede quando facciamo avanzare il punto base di un incrocio. #lemma[ @@ -1268,7 +1277,7 @@ Possiamo ora mostrare l'invarianza per scelta di punto base di $Omega_K (p)$. ) } - - Se $i$ è un sotto-incrocio in $K(p)$ allora $i$ sarà il primo incrocio nella sequenza di scambi ed avremo $i = n$ + - Se $i$ è un sotto-incrocio in $K(p)$ allora $i$ sarà il primo incrocio nella sequenza di scambi ed avremo $i = n$. La situazione sarà quindi la seguente: #{ set align(center) @@ -1347,7 +1356,7 @@ Possiamo ora mostrare l'invarianza per scelta di punto base di $Omega_K (p)$. e quindi $Omega_K (p) = Omega_K (q)$. - - Se $i$ è un sopra-incrocio in $K(p)$ siamo nella seguente situazione + - Se $i$ è un sopra-incrocio in $K(p)$ siamo nella seguente situazione: #{ set align(center) @@ -1362,7 +1371,7 @@ Possiamo ora mostrare l'invarianza per scelta di punto base di $Omega_K (p)$. ) } - 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 + 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) @@ -1377,21 +1386,30 @@ Possiamo ora mostrare l'invarianza per scelta di punto base di $Omega_K (p)$. ) } - 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 $q$: + Questa volta $i$ fa parte della sequenza di scambi per $K(q)$ poiché $i$ compare come ultimo incrocio nella visita e quindi in $hat(K)(q)$ sarà un sotto-incrocio. La situazione è più complessa di prima, $i$ comparirà come sopra-incrocio in mezzo alla sequenza di scambi. Avremo quindi le seguenti sequenze per $p$ e per $q$: + + $ + lambda(p) &= (n, n-1, dots, i+1, i-1, dots, 1, 0) \ + lambda(q) &= (n, n-1, dots, i+1, i, i-1, dots, 1, 0) \ + $ + + Se ora consideriamo gli scambi da applicare che ci portano a $hat(K)(p)$ e $hat(K)(q)$ abbiamo che $ - lambda(q) &= (n, n-1, dots, i, dots, 1, 0) \ - lambda(p) &= (n, n-1, dots, i+1, i-1, dots, 1, 0) + hat(K)(p) &= S_n S_(n-1) space.med dots.c space S_(i+1) S_(i-1) space.med dots.c space S_0 K \ + hat(K)(q) &= S_n S_(n-1) space.med dots.c space S_(i+1) S_i S_(i-1) space.med dots.c space S_0 K \ $ - a meno di riordinare gli scambi è facile vedere che $S_i hat(K)(q) = hat(K)(p)$, e che il @lemma-slide-identities si applica alla coppia $hat(K)(p), hat(K)(q)$ per l'indice $i$. Per il @lemma-sum-switches-rotation abbiamo anche 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. + 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 anche 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) \ lambda'(q) &= (i, i-1, dots, 1, 0, n, n-1, dots, i+1) $ - A questo punto possiamo applicare un argomento simile a quello del punto precedente. + A questo punto possiamo applicare lo stesso argomento del punto precedente. E questo completa la dimostrazione dell'indipendenza di $Omega_K (p)$ dalla scelta di punto base. ] @@ -1409,9 +1427,9 @@ A questo punto possiamo vedere che $L_K$ verifica gli assiomi della @kauffman-po #proof[ Mostriamo per induzione sul numero di incroci nel diagramma. - 1. Consideriamo i seguenti casi: #margin-note[ricontrollare tutta questa] + 1. Consideriamo i seguenti casi: - - 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 + - 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. Ora consideriamo l'espressione del caso @kauffman-rec-single-component, con una singola componente: $ L_K (a, z) &colon.eq @@ -1425,15 +1443,15 @@ A questo punto possiamo vedere che $L_K$ verifica gli assiomi della @kauffman-po ] $ - 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: + 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: $ => 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 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 di una componente con se stessa: in questo caso $i$ non compare in nessuna sequenza di rialzo in quanto le sequenze di rialzo scambiano solo incroci tra componenti diverse. Dunque possiamo applicare il caso @kauffman-rec-multi-component dell'induzione senza problemi. - - 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: + - 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, ricordiamo l'espressione in questione: #{ set text(size: small-size) @@ -1446,7 +1464,7 @@ A questo punto possiamo vedere che $L_K$ verifica gli assiomi della @kauffman-po $ } - 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. + Per quanto riguarda gli addendi 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 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. @@ -1462,33 +1480,127 @@ Come ultimo risultato vediamo che $L_K$ è un invariante di isotopia regolare. ] #proof[ - 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]). + Procediamo sempre per induzione sul numero di incroci e mostriamo l'invarianza per mosse che non ne aumentano il numero: - - Mossa II: Ci sono più casi in base a se la mossa riguarda una o più componenti + - Mossa II: Ci sono vari casi in base a se la mossa II riguarda una o più componenti: - Se la mossa è su una sola componente allora ci basta scegliere il punto base come segue - #todo[disegnino] + #figure(image("assets/move-2-base-points.jpg", width: 6cm)) - in questo modo i due incroci non compariranno nella sequenza di scambi. Tutti i termini di ii.a) saranno invarianti per mosse di tipo II e lo sarà anche $L_K$. + in questo modo i due incroci non compariranno nella sequenza di scambi. Tutti i termini di @kauffman-rec-single-component saranno invarianti per mosse di tipo II e lo sarà anche $L_K$. - Se la mossa riguarda più componenti allora il caso peggiore è il seguente - #todo[disegnino] + #figure(image("assets/move-2-base-points-bad.png", width: 3cm)) + + Vorremo vedere che vale $L[K] = L[S_2 S_1 K]$. Consideriamo le seguenti relazioni: + + $ + L[K] + L[S_1 K] &= z (L[E_1 K] + L[e_1 K]) \ + L[S_1 K] + L[S_2 S_1 K] &= z (L[E_2 S_1 K] + L[e_2 S_1 K]) \ + $ - Vediamo che $L[K] = L[S_2 S_1 K]$... + i diagrammi associati a queste relazioni sono riportati qui sotto, in particolare possiamo notare che ci sono alcune relazioni tra questi: + + #{ + set align(center) + set text(small-size) + + grid( + columns: 4, + column-gutter: 1em, + row-gutter: 0.5em, + image("assets/derived/atlas-move-2-K-0.png", width: 1.5cm), + image("assets/derived/atlas-move-2-K-1.png", width: 1.5cm), + image("assets/derived/atlas-move-2-K-2.png", width: 1.5cm), + image("assets/derived/atlas-move-2-K-3.png", width: 1.5cm), + $K$, + $S_1 K$, + $E_1 K$, + $e_1 K$, + ) + } + + #{ + set align(center) + set text(small-size) + + grid( + columns: 4, + column-gutter: 1em, + row-gutter: 0.5em, + image("assets/derived/atlas-move-2-S1K-0.png", width: 1.5cm), + image("assets/derived/atlas-move-2-S1K-1.png", width: 1.5cm), + image("assets/derived/atlas-move-2-S1K-2.png", width: 1.5cm), + image("assets/derived/atlas-move-2-S1K-3.png", width: 1.5cm), + $S_1 K$, + $S_2 S_1 K$, + $E_2 S_1 K$, + $e_2 S_1 K$, + ) + } + + #grid( + columns: 2, + column-gutter: 2em, + row-gutter: 0.5em, + // do not remove this comment (due to auto-formatter) + $ + #image("assets/derived/atlas-move-2-K-3.png", width: 1.25cm) + tilde + #image("assets/derived/atlas-move-2-S1K-3.png", width: 1.25cm) + $, + $ + #image("assets/derived/atlas-move-2-K-2.png", width: 1.25cm) + #h(8.1em) + #image("assets/derived/atlas-move-2-S1K-2.png", width: 1.25cm) + $, + $L[e_1 K] = L[e_2 S_1 K]$, + $L[E_1 K] = a^(-1) L[#skein.h] = L[E_2 S_1 K]$, + ) + + Dunque i due membri di destra di $(3)$ coincidono ed abbiamo quanto cercato: + + $ + L[K] + L[S_1 K] = z (L[E_1 K] + L[e_1 K]) = L[S_1 K] + L[S_2 S_1 K] \ + => L[K] + cancel(L[S_1 K]) = cancel(L[S_1 K]) + L[S_2 S_1 K] \ + => L[K] = L[S_2 S_1 K] + $ + + A questo punto nel diagramma $S_2 S_1 K$, gli incroci della mossa II non compariranno nella sequenza di rialzo dunque l'invarianza segue per induzione come prima. - Mossa III: - - 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: + - Se la mossa è su fili di una sola componente allora possiamo scegliere il punto base in modo che sia sul filo che passa sopra tra i tre, e saremo nella situazione seguente: + + $ + #image("assets/derived/atlas-move-3-0.png", height: 2.25cm) + quad + tilde + quad + #image("assets/move-3-result.png", height: 2.25cm) + $ + + In questo modo due dei tre incroci non compariranno nella sequenza di scambi, per quanto riguarda l'incrocio rimanente $*$ o già non compariva nella sequenza di scambi o possiamo procedere per induzione. Nel caso di splice abbiamo i seguenti casi + + #{ + set align(center) + set text(size: small-size) - #todo[disegnino] + grid( + columns: 2, + column-gutter: 2em, + image("assets/derived/atlas-move-3-2.png", height: 2.25cm), + image("assets/derived/atlas-move-3-3.png", height: 2.25cm), + ) + } - In un caso possiamo procedere per induzione, se invece ci sono degli splice possiamo utilizzare le seguenti equivalenze di diagrammi e poi procedere per induzione + Ma possiamo risolverli utilizzando equivalenze di diagrammi come le seguenti, in cui usiamo solo mosse di tipo II: - #todo[disegnino] + #figure(image("assets/atlas-move-3-equivalences.png", height: 2.25cm)) - - 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. + - 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 in modo che non ci siano due incroci consecutivi che appartengano ad una sequenza di rialzo, e procedere nuovamente per induzione. - Questo completa la dimostrazione. + Abbiamo così mostrato che per ogni diagramma di nodi e link, il polinomio $L_K$ è invariante per mosse di tipo III e mosse di tipo II che riducono il numero di incroci. Dunque è anche invariante per mosse di tipo II in entrambe le direzioni e questo completa la dimostrazione. ] \ No newline at end of file diff --git a/src/prelude.typ b/src/prelude.typ index 2947423..797b30c 100644 --- a/src/prelude.typ +++ b/src/prelude.typ @@ -147,4 +147,6 @@ } #h(0.125em)$ } -#let marker(content) = metadata(("marker", content)) \ No newline at end of file +#let marker(content) = metadata(("marker", content)) + +#let qquad = h(2em) \ No newline at end of file diff --git a/src/scripts/split-atlas.sh b/src/scripts/split-atlas.sh index 224f2a3..02b5b04 100755 --- a/src/scripts/split-atlas.sh +++ b/src/scripts/split-atlas.sh @@ -23,6 +23,8 @@ for i in $(seq 0 3); do X_OFFSET=$(( i * TILE_WIDTH )) TEMP_FILE="_temp_${i}.png" OUTPUT_FILE=$(printf "$OUTPUT_TEMPLATE" "$i") + + echo "Generating tile $i: $OUTPUT_FILE" magick "$INPUT_IMAGE" -crop "${TILE_WIDTH}x${TILE_HEIGHT}+${X_OFFSET}+0" +repage "$TEMP_FILE"

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