From c3e479bc782b3e83ce3e84b54f376a2d3075a52a Mon Sep 17 00:00:00 2001 From: Antonio De Lucreziis Date: Fri, 27 Jun 2025 00:28:42 +0200 Subject: [PATCH] more work, chapter 2 almost done --- out/tesi-triennale.pdf | Bin 1678656 -> 1639394 bytes src/assets/old/writhe-examples-old.png | Bin 0 -> 220232 bytes src/assets/old/writhe-examples.png | Bin 0 -> 220087 bytes src/assets/writhe-examples.png | Bin 220232 -> 171165 bytes src/main.typ | 34 +++++++++++++++++-------- 5 files changed, 23 insertions(+), 11 deletions(-) create mode 100644 src/assets/old/writhe-examples-old.png create mode 100644 src/assets/old/writhe-examples.png diff --git a/out/tesi-triennale.pdf b/out/tesi-triennale.pdf index d3597b096ecd0d78710359e4f24dbded1db9218c..f697c80a578ad66836493d92cb8d79327402db6d 100644 GIT binary patch delta 154292 zcmZ5{1yojDx3x$JNJw{x2-4CeA*D!3ceiw>92zC1yQMp&JEXfCB&56J-`@Xz-|KrB zoB^Km?6db?d#*X>S`1t2j8+2-N(bRkIJnr!xmo^vvvEhx!Z|(WdL8+(gTEpi{s;zY 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Nel caso del polinomio $F_K$, il calcolo 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. -Questo sarà importante più avanti quando vedremo invarianti come il polinomio $F_K$ che risulteranno essere invarianti per isotopia ambiente. Questi, però, sono definiti attraverso l'isotopia regolare; in particolare il calcolo va fatto fissando un particolare diagramma del link. Ad esempio è tutt'ora un problema aperto dare una definizione di $F_K$ che sia indipendente dalla scelta di un diagramma. +// 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. == Writhe -Vediamo ora un primo invariante di isotopia regolare per link orientatati, prima definiamo il segno di un incrocio. +Vediamo ora un primo invariante di isotopia regolare per link orientatati, per prima cosa definiamo il segno di un incrocio. #definition[ - Dato un incrocio di un link orientato ci possono essere due casi in base alla direzione in cui il filo sottostante si trova rispetto a quello sopra. Definiamo *il segno* di un incrocio in base a questi due casi: + Dato un incrocio di un link orientato ci possono essere due casi in base alla direzione in cui il filo sottostante si trova rispetto a quello sopra. Definiamo *il segno* di un incrocio in base a questi casi come segue $ epsilon(#skein-generic(direction: (+1, +1))) = +1 @@ -303,7 +315,7 @@ Vediamo ora un primo invariante di isotopia regolare per link orientatati, prima ] #definition[ - Dato $K$ link orientato, il *writhe* $w(K)$ (o numero di avvolgimento) è dato dalla somma dei segni dei suoi incroci + Dato $K$ link orientato, il *writhe* $w(K)$ o numero di avvolgimento è dato dalla somma dei segni dei suoi incroci $ w(K) colon.eq sum_c epsilon(c) @@ -348,7 +360,7 @@ Questo risultato può essere generalizzato a link con l'accortezza di invertire ] #figure( - image("assets/writhe-examples.png", width: 14cm), + image("assets/writhe-examples.png", width: 10cm), caption: [ Esempi di calcolo del writhe // per varie orientazioni del link di Hopf e del nodo trifoglio @@ -535,7 +547,7 @@ $ $ -Ed del nodo trifoglio +E del nodo trifoglio // #figure(image("assets/implicit-calc-3.png"))
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