From 195d3b21c5702911610898a074e5afd0c2f251c1 Mon Sep 17 00:00:00 2001 From: Antonio De Lucreziis Date: Fri, 13 Jun 2025 16:58:11 +0200 Subject: [PATCH] chapter 1 almost over --- makefile | 11 ++ out/tesi-triennale.pdf | Bin 241610 -> 275985 bytes out/tesi-triennale@capitolo-1.pdf | Bin 0 -> 240854 bytes .../ambient-regular-isotopy-difference.png | Bin 0 -> 49417 bytes src/main.typ | 103 +++++++++--------- src/theme.typ | 2 +- 6 files changed, 66 insertions(+), 50 deletions(-) create mode 100644 makefile create mode 100644 out/tesi-triennale@capitolo-1.pdf create mode 100644 src/assets/ambient-regular-isotopy-difference.png diff --git a/makefile b/makefile new file mode 100644 index 0000000..07e640e --- /dev/null +++ b/makefile @@ -0,0 +1,11 @@ + + + +.PHONY: capitolo-1 +capitolo-1: out/tesi-triennale@capitolo-1.pdf + +out/tesi-triennale.pdf: src/main.typ + typst compile $< $@ + +out/tesi-triennale@capitolo-1.pdf: out/tesi-triennale.pdf + pdftk $< cat 1-7 output $@ \ No newline at end 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zKb&mf_mZoo28l$nPZ8i3^%SrB)KUK9AOGhcx6dz3Pxka!Y>a!3(@GS9TLaGc?|lDn D_>yUa literal 0 HcmV?d00001 diff --git a/src/main.typ b/src/main.typ index 1f55779..1c99414 100644 --- a/src/main.typ +++ b/src/main.typ @@ -22,8 +22,8 @@ #show: ams-article.with( paper-size: "a4", - title: [Il polinomio di Kauffman, \ un Invariante di Isotopia Regolare], - page-title: [Il polinomio di Kauffman, un Invariante di Isotopia Regolare], + title: [Il polinomio di Kauffman: \ un invariante di isotopia regolare], + page-title: [Il polinomio di Kauffman: un invariante di isotopia regolare], authors: ( ( name: "Antonio De Lucreziis", @@ -33,7 +33,7 @@ ), ), abstract: [ - In questa tesi tratteremo del polinomio di Kauffman, un invariante di nodi e link a meno di isotopia regolare. Introdurremo dei risultati di fondazione della teoria dei nodi per definire l'isotopia regolare. Infine vedremo la dimostrazione della buona definizione del polinomio di Kauffman a partire dalla forma assiomatica. Inoltre per il progetto di Laboratorio Computazionale abbiamo implementato in Python questo polinomio e verificato tutti i valori presenti nel database di KnotInfo e trovato un errore nel nodo $10_125$. + In questa tesi tratteremo del polinomio di Kauffman, un invariante di nodi e link per di isotopia regolare. Introdurremo dei risultati di fondazione della teoria dei nodi per definire l'isotopia regolare. Infine vedremo la dimostrazione della buona definizione del polinomio di Kauffman a partire dalla forma assiomatica. Inoltre per il progetto di Laboratorio Computazionale abbiamo implementato in Python questo polinomio e verificato tutti i valori presenti nel database di KnotInfo e trovato un errore nel calcolo del nodo $10_125$. ], bibliography: bibliography("refs.bib"), ) @@ -44,9 +44,11 @@ = Introduzione -In questa tesi studieremo un invariante di isotopia regolare chiamato polinomio di Kauffman @Kauffman1990-qe. Intuitivamente vedremo che se l'isotopia ambiente è l'equivalenza tra diagrammi generata dalle mosse I, II e III di Reidemeister, l'isotopia regolare è l'equivalenza generata solo dalle mosse II e III. +In questa tesi studieremo un invariante di isotopia regolare chiamato polinomio di Kauffman @Kauffman1990-qe. Intuitivamente se l'isotopia ambiente è l'equivalenza tra diagrammi generata dalle mosse I, II e III di Reidemeister, l'isotopia regolare è l'equivalenza generata solo dalle mosse II e III. #margin-note[Il disegno è da rifare] -Dato un nodo o link non orientato $K$, possiamo definire in forma assiomatica il polinomio $kL_K (a, z) in bb(Z)[a, a^(-1), z, z^(-1)]$ attraverso le seguenti relazioni: +#figure(image("assets/ambient-regular-isotopy-difference.png", width: 100%)) + +Dato un nodo o link non orientato $K$, possiamo definire in forma assiomatica il polinomio $kL_K (a, z) in bb(Z)[a, a^(-1), z, z^(-1)]$ attraverso i seguenti assiomi: 1. Se $K$, $K'$ sono equivalenti a meno di isotopia regolare, allora $kL_K (a,z) = kL_K' (a,z)$. @@ -61,7 +63,7 @@ Dato un nodo o link non orientato $K$, possiamo definire in forma assiomatica il - $kL(#skein.under-twist) = a^(-1) kL(#skein.strand)$ -Vedremo come $kL_K (a, z)$ è ben definito ed è un invariante di isotopia regolare e come può essere generalizzato a un invariante di isotopia ambiente. +Vedremo come questi assiomi definiscono un unico invariante di isotopia regolare ben definito e come $kL_K (a, z)$ può essere generalizzato a un invariante di isotopia ambiente $F_K (a, z)$ utilizzando il writhe di un nodo. == Introduzione alla Teoria dei Nodi @@ -74,52 +76,52 @@ Ora introdurremo alcuni risultati fondamentali di teoria dei nodi tra cui le def // #let embedding-def = ref-link()[embedding] #fact[ - Dati due spazi topologici con $X$ compatto, $Y$ di Hausdorff ed $f : X arrow Y$ continua allora $f$ è un embedding $<=>$ è iniettiva. + Dati $X$, $Y$ spazi topologici con $X$ compatto, $Y$ di Hausdorff e data un'applicazione continua $f : X arrow Y$ allora $f$ è un embedding $<=>$ è iniettiva. ] #definition[ Dato un embedding $f : X arrow Y$, un punto $p in X$ allora $f$ si dice *localmente piatto* in $p$ se $exists U subset bb(R)^3$ intorno di $p$, tale che +] - #align(center, image("assets/locally-flat.jpg", width: 50%)) - // #align( - // center, - // cetz.canvas( - // { - // import cetz.draw: * - +#{ + set align(center) + grid( + columns: 3, + gutter: 1em, + align: center, + $U$, $approx$, $bb(D)^2 times [0, 1]$, + $U inter K$, $<->$, ${ 0 } times [0, 1]$, + ) +} + +#align(center, image("assets/locally-flat.jpg", width: 35%)) +// #align( +// center, +// cetz.canvas( +// { +// import cetz.draw: * - // hobby((0, 1), (0.6, 0.6), (1, 0), omega: 2, name: "l") - // content((name: "l", anchor: 33%), box(fill: white, $x$)) +// hobby((0, 1), (0.6, 0.6), (1, 0), omega: 2, name: "l") - // circle((name: "l", anchor: 33%), radius: (2em, 3em)) +// content((name: "l", anchor: 33%), box(fill: white, $x$)) +// circle((name: "l", anchor: 33%), radius: (2em, 3em)) - // // let angle = cetz-path-angle("l", 33%) +// // let angle = cetz-path-angle("l", 33%) - // let result = cetz.path-util.direction((line((0, 1), (0.6, 0.6), (1, 0)),), 33%) - // content((1, 1), [#result]) - // }, +// let result = cetz.path-util.direction((line((0, 1), (0.6, 0.6), (1, 0)),), 33%) - // length: 33%, - // ), - // ) +// content((1, 1), [#result]) +// }, - #{ - set align(center) - grid( - columns: 3, - gutter: 1em, - align: center, - $U$, $approx$, $bb(D)^2 times [0, 1]$, - $U inter K$, $<->$, ${ 0 } times [0, 1]$, - ) - } +// length: 33%, +// ), +// ) - inoltre $f$ si dice *localmente piatto* (ovunque) se lo è in ogni punto di $X$. -] +inoltre $f$ si dice *localmente piatto* (ovunque) se lo è in ogni punto di $X$. // #let locally-flat-def = ref-link()[localmente piatto] @@ -127,8 +129,6 @@ Ora introdurremo alcuni risultati fondamentali di teoria dei nodi tra cui le def Un *nodo tame* è un sottoinsieme $K subset bb(R)^3$ per cui esiste un embedding $f : bb(S)^1 arrow.hook bb(R)^3$ localmente piatto con $K = f(bb(S)^1)$. In questo caso $f$ è anche detto *embedding tame*. ] -Esistono anche nodi non tame come il seguente, ma d'ora in avanti considereremo solo nodi tame.#margin-note[aggiungere citazione a wikipedia per l'immagine sotto?] - #figure( image("assets/wild_knot.svg", width: 75%), caption: [ @@ -136,12 +136,14 @@ Esistono anche nodi non tame come il seguente, ma d'ora in avanti considereremo ], ) +Esistono anche nodi non tame come il precedente, ma d'ora in avanti considereremo solo nodi tame. #margin-note[l'immagine sotto è presa da wikipedia, aggiungere una citazione?] + // #todo[ // Disegno nodo non tame // ] #definition[ - Due nodi $K_0, K_1 subset bb(R)^3$ sono *equivalenti* se esiste un'*isotopia ambiente* $H$ che porta uno nell'altro, ovvero esiste un'applicazione continua $H : bb(R)^3 times [0, 1] arrow bb(R)^3$, tale che + Due nodi $K_0, K_1 subset bb(R)^3$ sono *equivalenti* se esiste un'*isotopia ambiente* $H$ che porta uno nell'altro, ovvero esiste $H : bb(R)^3 times [0, 1] arrow bb(R)^3$ applicazione continua, tale che - $forall t in [0, 1], H(dot, t)$ è un omeomorfismo @@ -154,7 +156,7 @@ Esistono anche nodi non tame come il seguente, ma d'ora in avanti considereremo Inoltre se due nodi sono equivalenti allora vale che $bb(R)^3 without K_0 approx bb(R)^3 without K_1$. -Spesso serve fare manipolazioni direttamente sui nodi o sui diagrammi e lavorare con gli embedding $bb(S)^1 arrow bb(R)^3$ non è molto comodo da un punto di vista pratico, per questo motivo introduciamo i _nodi poligonali_ che ci permettono di dare una formalizzazione alternativa dei nodi tame. +Spesso serve fare manipolazioni direttamente sui nodi o sui diagrammi e lavorare con gli embedding $bb(S)^1 arrow bb(R)^3$ non è molto comodo da un punto di vista formale, per questo motivo introduciamo i _nodi poligonali_ che ci permettono di dare una formalizzazione alternativa dei nodi tame più operativa. #definition[ Un *nodo poligonale* è un nodo equivalente ad un'unione finita di segmenti lineari. @@ -170,12 +172,15 @@ Spesso serve fare manipolazioni direttamente sui nodi o sui diagrammi e lavorare Il *nodo banale* è la classe di equivalenza del bordo di un triangolo equilatero. ] -Possiamo generalizzare tutte le definizioni che abbiamo appena dato da nodi a link, sostituendo $bb(S)^1$ con $bb(S)^1 union dots.c union bb(S)^1$ con le opportune modifiche, ad esempio la definizione di link è la seguente: +Passiamo ora al caso in cui abbiamo un nodo con più di una componente #definition[ $L subset bb(R)^3$ è detto *link* se $exists$ embedding $f : bb(S)^1 union dots.c union bb(S)^1 arrow bb(R)^3$ localmente piatto con immagine $L$. ] + +Possiamo generalizzare tutte le definizioni che diamo per i nodi ai link, sostituendo $bb(S)^1$ con $bb(S)^1 union dots.c union bb(S)^1$ con le opportune modifiche. + Introduciamo ora il concetto di equivalenza combinatoria di link poligonali in $bb(R)^3$, questo è il primo passo che ci permette di descrivere l'equivalenza tra nodi attraverso sequenza finita di mosse. #definition[ @@ -185,7 +190,7 @@ Introduciamo ora il concetto di equivalenza combinatoria di link poligonali in $ #todo[disegno 1] - 2. Dato un triangolo piano $Delta subset bb(R)^3$ tale che $Delta inter L$ sia un lato di $Delta$ ed un segmento di $L$ allora una _$Delta$-move_ è la seguente + 2. Dato un triangolo piano $Delta subset bb(R)^3$ tale che $Delta inter L$ sia un lato di $Delta$ ed un segmento di $L$ allora una $Delta$*-mossa* è la seguente #todo[disegno 2] ] @@ -200,7 +205,7 @@ Data una direzione $v in bb(S)^2 subset bb(R)^3$, possiamo definire $pi_v : bb(R #todo[disegno] -Per formalizzare il concetto di diagramma di un nodo introdurremo il concetto di proiezione regolare, ovvero una proiezione in cui i punti di intersezione tra le immagini dei segmenti di $L$ sono solamente gli incroci del nodo (e senza segmenti paralleli alla direzione di proiezione). +Per formalizzare il concetto di diagramma di un nodo introdurremo il concetto di proiezione regolare, ovvero una proiezione in cui i punti di intersezione tra le immagini dei segmenti di $L$ sono solamente gli incroci (punti doppi) del nodo (e senza segmenti paralleli alla direzione di proiezione). #definition[ Sia $L subset bb(R)^3$ un link, $v in bb(S)^2$ una direzione e $pi_v : bb(R)^3 arrow v^perp$ la proiezione su $v^perp$ come in precedenza. Allora un punto $x in pi(L) subset v^perp$ si dice @@ -229,11 +234,11 @@ Per formalizzare il concetto di diagramma di un nodo introdurremo il concetto di Quindi dato $L subset bb(R)^3$ link poligonale, esiste una proiezione regolare $pi(L) subset bb(R)^2$, ovvero una con un numero finito di punti singolari, ciascuno dei quali è un punto doppio e di intersezione trasversa. Ognuno di questi punti doppi viene detto *incrocio*. #definition[ - Un *diagramma* $D subset bb(R)^2$ di un link $L subset bb(R)^3$ è una proiezione regolare di $L$ decorata con l'informazione sopra/sotto ad ogni incrocio. + Un *diagramma* $D subset bb(R)^2$ di un link $L subset bb(R)^3$ è l'immagine di $L$ attraverso una proiezione regolare decorata con l'informazione sopra/sotto ad ogni incrocio. ] #fact[ - Due link con lo stesso diagramma sono equivalenti. + Due link con stesso diagramma sono equivalenti. ] // #todo[ @@ -261,17 +266,17 @@ Come abbiamo generalizzato nodi a link possiamo ripetere tutto anche aggiungendo #definition[ Dato un link $L subset bb(R)^3$ possiamo definire - - Il *mirror* $m(L) colon.eq rho(L)$ dove $rho : bb(R)^3 arrow bb(R)^3$ è la riflessione rispetto al piano di proiezione. In particolare dato un diagramma è lo stesso diagramma con l'informazione sopra/sotto scambiata. + - Il *mirror* $m(L) colon.eq rho(L)$ dove $rho : bb(R)^3 arrow bb(R)^3$ è la riflessione rispetto al piano di proiezione. In particolare dato un diagramma, il suo mirror è lo stesso diagramma con tutte le informazioni sopra/sotto scambiate. - e se $L$ è anche _orientato_ allora possiamo definire + inoltre, se diamo anche un'_orientazione_ ad $L$ possiamo definire - - Il *reverse* $r(L)$ che è lo stesso $L$ con l'orientazione opposta su ogni componente. + - Il *reverse* $r(L)$, in cui prendiamo l'orientazione opposta su ogni componente. - L'*inverso* $-L colon.eq m(r(L)) = r(m(L))$. ] #definition[ - Se $K subset bb(R)^3$ e $K tilde.not m(K)$ allora diciamo che $K$ è *chirale*, altrimenti è detto *anfichirale*. + Se $K subset bb(R)^3$ e $K$ non è equivalente a $m(K)$ allora diciamo che $K$ è *chirale*, altrimenti è detto *anfichirale*. ] // == Relazioni Skein diff --git a/src/theme.typ b/src/theme.typ index 9751f8a..7b49b16 100644 --- a/src/theme.typ +++ b/src/theme.typ @@ -416,7 +416,7 @@ set text(size: heading-level-size(2)) block(it.body) } - heading(level: 2)[Bibliografia] + heading(level: 1)[Bibliografia] } bibliography