From 5c1df0c0e870cf7f2485ed86dee8328e95aa518d Mon Sep 17 00:00:00 2001 From: Antonio De Lucreziis Date: Thu, 19 Jun 2025 18:23:04 +0200 Subject: [PATCH] ultimi ritocchi primo capitolo --- out/tesi-triennale.pdf | Bin 1009723 -> 695658 bytes src/assets/reidemeister-moves.jpg | Bin 0 -> 233692 bytes src/backup.typ | 59 +++ src/main.typ | 763 ++++++++++++++---------------- src/prelude.typ | 19 +- src/theme.typ | 5 +- 6 files changed, 435 insertions(+), 411 deletions(-) create mode 100644 src/assets/reidemeister-moves.jpg create mode 100644 src/backup.typ diff --git a/out/tesi-triennale.pdf b/out/tesi-triennale.pdf index 0331d7d364c56e13247961799a820cc4f60761d7..195bc97d8b26a00fa17a24e36a628dcf89498fe7 100644 GIT binary patch delta 137297 zcmY(q18^l>7qC0AZQGgPL?^aA@dOiF6Z-@c+qP}nwrwX9TX){?|LfkmT~Akauf6x` zUAwxgySmoXKEr?3MvqRVBqs5lm5B?U%J<&}56{Y(Tq})?5qCueC8;W^DHp3K3(m?F zzdMA1$;wU2Liz(_8UM{{sL3Bn7hm2OBpxDG=}<|7&F<1+tN{ zv2u_C+5dx!^IzcK2?r?$=f42we*jqiW#Rgd|Bd*U=-)M1IGDJ(xk23j;@qU%ETrtL zKvHhj|KQ+YA>{`A2OImpBews+%1O%2{vVwGQgiqoUP5&};|C

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--git a/src/backup.typ b/src/backup.typ new file mode 100644 index 0000000..21c59cb --- /dev/null +++ b/src/backup.typ @@ -0,0 +1,59 @@ + +// == Relazioni Skein + +// // Note. Here and elsewhere in this paper, small diagrams stand for parts of larger +// // diagrams. A collection of small diagrams occuring in a single equation all share +// // the same larger diagram. (The large diagram is only changed as indicated by +// // the small diagrams.) + +// Prima di passare alla definizione dell'isotopia regolare, introduciamo brevemente le relazioni skein. Sono uno strumento utile per definire molti invarianti costruiti su diagrammi. L'idea è di rappresentare relazioni tra più diagrammi indicando con dei diagrammi parziali solo le parti in cui questi differiscono. + + +// #definition[ +// Diamo un nome alle segmenti porzioni di diagrammi +// ] + +// #align( +// center, +// grid( +// columns: 4, +// column-gutter: 2em, +// row-gutter: 1em, +// skein-generic(kind: "over", arrows: (false, false)), +// skein-generic(kind: "under", arrows: (false, false)), +// skein.h, +// skein.v, + +// $L_+$, $L_-$, $L_0$, $L_infinity$, +// ), +// ) + +// I primi due indicano uno stesso diagramma in cui abbiamo scambiato un incrocio, le seconde invece sono chiamate da Kauffman _splice_ e rappresentano un incrocio in cui abbiamo tagliato e rincollato i due fili. Queste ultime, in letteratura, sono anche chiamate spesso _smoothing_ o risoluzioni.#margin-note[come tradurre questi termini in italiano?] + +// Fissato un anello $scr(R)$, un possibile modo di formalizzare le relazioni skein è di vederle come elementi dell'anello di polinomi $scr(R)[L_+, L_-, L_0, L_infinity]$, ad esempio preso un elemento $F in scr(R)[L_+, L_-, L_0, L_infinity]$, una relazione skein è un'espressione della forma + +// $ +// F(L_+, L_-, L_0, L_infinity) = 0 +// $ + +// #definition[ +// Fissato un diagramma $D$ ed un suo incrocio $c$ possiamo definire le seguenti operazioni sul diagramma + +// - $S_c K$ è lo stesso diagramma $D$ con l'incrocio $c$ scambiato, ovvero con l'informazione sopra/sotto invertita. + +// - $E_c K$ è il diagramma con l'incrocio $c$ sostituito con lo _splice_ orizzontale + +// - $e_c K$ è il diagramma con l'incrocio $c$ sostituito con lo _splice_ verticale +// ] + +// A priori non è ovvio che gli _splice_ siano ben definiti e potrebbero dipendere dall'orientazione del diagramma [...]. + +// // Però si può dimostrare che c'è un modo indipendente dall'orientazione del nodo di decidere come applicare i due tipi di splice in modo coerente. + +// A questo punto possiamo introdurre un concetto di valutazione di queste relazioni skein, data una relazione $F(L_+, L_-, L_0, L_infinity) = 0$ possiamo introdurre la seguente valutazione. Fissato diagramma $D$ ed un suo incrocio positivo $c$ possiamo definire $F_D in scr(R)[scr(D)]$ come #margin-note[Trovare referenze per questa parte anche se probabilmente va trattata diversamente o tolta e basta forse] + +// $ +// F_D colon.eq F(space D , space S_c D , space E_c D , space e_c D space) +// $ + +// // Questo ci permette di valutare una relazione skein "vicino" ad un incrocio specifico. diff --git a/src/main.typ b/src/main.typ index 1413331..981abf9 100644 --- a/src/main.typ +++ b/src/main.typ @@ -44,10 +44,10 @@ = Introduzione -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] +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. #figure( - image("assets/ambient-regular-isotopy-difference.jpg", width: 100%), + image("assets/ambient-regular-isotopy-difference.jpg", width: 13cm), caption: [Mosse di Reidemeister], ) @@ -55,7 +55,7 @@ Dato un nodo o link non orientato $K$, possiamo definire in forma assiomatica il 1. Se $K$, $K'$ sono equivalenti a meno di isotopia regolare, allora $kL_K (a,z) = kL_K' (a,z)$. -2. Valgono le seguenti relazioni skein +2. Valgono le seguenti relazioni skein: - $kL(#skein.over) + kL(#skein.under) = z (kL(#skein.h) + kL(#skein.v))$ @@ -66,11 +66,11 @@ 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 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. +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 -Ora introdurremo alcuni risultati fondamentali di teoria dei nodi tra cui le definizioni di nodo e link ed alcuni risultati di fondazione che ci serviranno per poter parlare ad esempio dei diagrammi planari e dell'isotopia regolare. +Ora introdurremo alcuni risultati fondamentali di teoria dei nodi. Vedremo le definizioni di nodo e link ed alcuni risultati di fondazione che ci serviranno per poter parlare ad esempio dei diagrammi planari e dell'isotopia regolare. #definition[ Dati $X, Y$ spazi topologici, $f : X arrow Y$ continua si dice *embedding* se $X$ è omeomorfo a $f(X) subset Y$ con la topologia di sottospazio indotta da $Y$. @@ -97,7 +97,7 @@ Ora introdurremo alcuni risultati fondamentali di teoria dei nodi tra cui le def $U$, $approx$, $bb(D)^2 times [0, 1]$, $U inter f(X)$, $<->$, ${ 0 } times [0, 1]$, ), - align(center, image("assets/locally-flat-v2.png", width: 6cm)), + align(center, image("assets/locally-flat-v2.png", width: 5.5cm)), ) @@ -118,16 +118,12 @@ inoltre $f$ si dice embedding *localmente piatto* se lo è in ogni punto di $X$. Esistono anche nodi non tame come quello riportato nella figura precedente @wiki:wild_knot, ma d'ora in avanti considereremo solo nodi tame. -// #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 $H : bb(R)^3 times [0, 1] arrow bb(R)^3$ applicazione continua, tale che - $forall t in [0, 1], H(dot, t)$ è un omeomorfismo - e ponendo $H_t (x) colon.eq H(x, t)$: + e ponendo $H_t (x) colon.eq H(x, t)$ si ha - $H_0 = id_(bb(R)^3)$ @@ -136,7 +132,7 @@ Esistono anche nodi non tame come quello riportato nella figura precedente @wiki 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 formale, per questo motivo introduciamo i _nodi poligonali_ che ci permettono di dare una formalizzazione alternativa dei nodi tame più operativa. +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 e pratico, 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. @@ -159,7 +155,7 @@ Passiamo ora al caso in cui abbiamo un nodo con più di una componente ] -Possiamo generalizzare tutte le definizioni che diamo per i nodi ai link, sostituendo $bb(S)^1$ con $bb(S)^1 union dots.c union bb(S)^1$ apportando le giuste modifiche. +Possiamo generalizzare tutte le definizioni che diamo per i nodi ai link, sostituendo $bb(S)^1$ con $bb(S)^1 union dots.c union bb(S)^1$ e apportando le giuste modifiche. Introduciamo ora il concetto di equivalenza combinatoria di link poligonali in $bb(R)^3$; questo è il primo passo che ci permetterà di descrivere l'equivalenza per isotopia ambiente tra nodi attraverso una sequenza finita di mosse. @@ -179,13 +175,15 @@ Introduciamo ora il concetto di equivalenza combinatoria di link poligonali in $ Due link poligonali $L_1, L_2 subset bb(R)^3$ sono equivalenti $<=>$ sono combinatorialmente equivalenti. ] +#pagebreak() + == Proiezioni e Diagrammi -Data una direzione $v in bb(S)^2 subset bb(R)^3$, possiamo definire $pi_v : bb(R)^3 arrow v^perp$ proiezione sul piano ortogonale a $v$ come segue +Data una direzione $v in bb(S)^2 subset bb(R)^3$, possiamo definire $pi_v : bb(R)^3 arrow v^perp$ proiezione sul piano ortogonale a $v$ come nella seguente figura: #figure(image("assets/projection-plane.png", width: 60%)) -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). +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). #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 @@ -198,7 +196,7 @@ Per formalizzare il concetto di diagramma di un nodo introdurremo il concetto di ] #fact[ - Dato un link poligonale $L subset bb(R)^3$, esiste un aperto denso $U subset bb(S)^2$ tale che $forall v in U$, $pi_v (L)$ soddisfa + Dato un link poligonale $L subset bb(R)^3$, esiste un aperto denso $U subset bb(S)^2$ tale che $forall v in U$ si ha che 1. $L$ non ha segmenti paralleli a $v$. @@ -208,10 +206,10 @@ Per formalizzare il concetto di diagramma di un nodo introdurremo il concetto di - $x$ è un punto doppio - - $x$ è di intersezione trasversa delle immagini di due segmenti + - $x$ è un punto di intersezione trasversa delle immagini di due segmenti ] -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*. +Quindi dato $L subset bb(R)^3$ link poligonale, esiste una proiezione regolare $pi(L) subset bb(R)^2$ 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 l'immagine di $L$ attraverso una proiezione regolare decorata con l'informazione sopra/sotto ad ogni incrocio. @@ -221,15 +219,13 @@ Quindi dato $L subset bb(R)^3$ link poligonale, esiste una proiezione regolare $ Due link con stesso diagramma sono equivalenti. ] -// #todo[ -// Vari commenti -// ] - #definition[ - L'insieme di isotopie planari e le mosse I, II, III (e loro inverse) sono le *mosse di Reidemeister*. + Le mosse I, II, III (e le loro inverse) sono dette *mosse di Reidemeister*. ] -#theorem(name: [Reidemeister])[ +#figure(image("assets/reidemeister-moves.jpg", width: 10cm)) + +#theorem(name: [di Reidemeister])[ Due diagrammi di link equivalenti sono collegati da una successione finita di isotopie planari e mosse di Reidemeister. ] @@ -240,7 +236,7 @@ Il modo principale in cui utilizzeremo il teorema di Reidemeister è attraverso ] #proof[ - È la forma contronominale del teorema di Reidemeister. + È semplicemente la forma contronominale del teorema di Reidemeister. ] Funzioni con la stessa proprietà di $phi$ sono quindi invarianti di nodi o link per isotopia ambiente e ci danno un modo di distinguerli. Più avanti vedremo ad esempio il polinomio $F_K (a, z) : scr(D) arrow bb(Z)[a, a^(-1), z, z^(-1)]$ che risulterà essere un invariante di isotopia ambiente per link orientatati. @@ -262,491 +258,452 @@ Come abbiamo generalizzato da nodi a link possiamo ripetere tutto anche aggiunge ] #definition[ - Se $K subset bb(R)^3$ e $K$ non è equivalente a $m(K)$ allora diciamo che $K$ è *chirale*, altrimenti è detto *anfichirale*. + Sia $K subset bb(R)^3$, se $K$ non è equivalente a $m(K)$ allora diciamo che $K$ è *chirale*, altrimenti è detto *anfichirale*. ] -// == Relazioni Skein - -// // Note. Here and elsewhere in this paper, small diagrams stand for parts of larger -// // diagrams. A collection of small diagrams occuring in a single equation all share -// // the same larger diagram. (The large diagram is only changed as indicated by -// // the small diagrams.) -// Prima di passare alla definizione dell'isotopia regolare, introduciamo brevemente le relazioni skein. Sono uno strumento utile per definire molti invarianti costruiti su diagrammi. L'idea è di rappresentare relazioni tra più diagrammi indicando con dei diagrammi parziali solo le parti in cui questi differiscono. += Isotopia Regolare +#todo[ + Work in progress +] -// #definition[ -// Diamo un nome alle segmenti porzioni di diagrammi +// #todo[ +// Aggiungere più commenti // ] -// #align( -// center, -// grid( -// columns: 4, -// column-gutter: 2em, -// row-gutter: 1em, -// skein-generic(kind: "over", arrows: (false, false)), -// skein-generic(kind: "under", arrows: (false, false)), -// skein.h, -// skein.v, - -// $L_+$, $L_-$, $L_0$, $L_infinity$, -// ), -// ) - -// I primi due indicano uno stesso diagramma in cui abbiamo scambiato un incrocio, le seconde invece sono chiamate da Kauffman _splice_ e rappresentano un incrocio in cui abbiamo tagliato e rincollato i due fili. Queste ultime, in letteratura, sono anche chiamate spesso _smoothing_ o risoluzioni.#margin-note[come tradurre questi termini in italiano?] +// Il teorema di Reidemeister ci dice che se due nodi sono equivalenti a meno di isotopia allora lo sono anche i loro diagrammi a meno di isotopie planari e delle mosse I, II e III. -// Fissato un anello $scr(R)$, un possibile modo di formalizzare le relazioni skein è di vederle come elementi dell'anello di polinomi $scr(R)[L_+, L_-, L_0, L_infinity]$, ad esempio preso un elemento $F in scr(R)[L_+, L_-, L_0, L_infinity]$, una relazione skein è un'espressione della forma - -// $ -// F(L_+, L_-, L_0, L_infinity) = 0 -// $ +// Possiamo chiederci cosa succede se ci restringiamo all'equivalenza generata solo dalle isotopie planari e dalle mosse II e III, questo ci dà una nuova equivalenza tra nodi più forte dell'isotopia ambiente. // #definition[ -// Fissato un diagramma $D$ ed un suo incrocio $c$ possiamo definire le seguenti operazioni sul diagramma +// Due nodi o link $K_1, K_2$ si dicono equivalenti a meno di *isotopia regolare* se due loro diagrammi sono equivalenti a meno di isotopie planari e mosse II e III di Reidemeister. +// ] -// - $S_c K$ è lo stesso diagramma $D$ con l'incrocio $c$ scambiato, ovvero con l'informazione sopra/sotto invertita. +// #figure(image("assets/ambient-regular-isotopy-difference.jpg", width: 13cm)) -// - $E_c K$ è il diagramma con l'incrocio $c$ sostituito con lo _splice_ orizzontale +// Una prima cosa che possiamo notare è che tutte le mosse R1 possono essere "fattorizzate" in modo da averle tutte raggruppate insieme. #margin-note[Magari aggiungere la dimostrazione di questa cosa] -// - $e_c K$ è il diagramma con l'incrocio $c$ sostituito con lo _splice_ verticale +// #todo[ +// Altro disegno // ] -// A priori non è ovvio che gli _splice_ siano ben definiti e potrebbero dipendere dall'orientazione del diagramma [...]. -// // Però si può dimostrare che c'è un modo indipendente dall'orientazione del nodo di decidere come applicare i due tipi di splice in modo coerente. +// C'è un'osservazione importante da fare quando parliamo di isotopia regolare. Quest'ultima è definita solo quando si parla di diagrammi di link, quando invece parliamo di isotopia ambiente ha anche senso parlare dei link come circonferenze embedded. -// A questo punto possiamo introdurre un concetto di valutazione di queste relazioni skein, data una relazione $F(L_+, L_-, L_0, L_infinity) = 0$ possiamo introdurre la seguente valutazione. Fissato diagramma $D$ ed un suo incrocio positivo $c$ possiamo definire $F_D in scr(R)[scr(D)]$ come #margin-note[Trovare referenze per questa parte anche se probabilmente va trattata diversamente o tolta e basta forse] +// Più avanti vedremo invarianti di isotopia ambiente come il polinomio $F_K$ che però è definito attraverso l'isotopia regolare, in particolare il calcolo va fatto fissando un particolare diagramma del link. Tutt'ora è un problema aperto dare una definizione di $F_K$ che sia indipendente dalla scelta di un diagramma. -// $ -// F_D colon.eq F(space D , space S_c D , space E_c D , space e_c D space) -// $ -// // Questo ci permette di valutare una relazione skein "vicino" ad un incrocio specifico. -= Isotopia Regolare +// Questo ci suggerisce un primo invariante di isotopia regolare per link orientatati, per prima cosa introduciamo il segno di un incrocio. -#todo[ - Aggiungere più commenti -] +// #definition[ +// Dato un incrocio di un link orientato ci possono essere due casi -Il teorema di Reidemeister ci dice che se due nodi sono equivalenti a meno di isotopia allora lo sono anche i loro diagrammi a meno di isotopie planari e delle mosse I, II e III. +// $ +// epsilon(#skein-generic(direction: (+1, +1))) = +1 +// #h(2em) +// epsilon(#skein-generic(direction: (+1, -1))) = -1 +// $ +// ] -Possiamo chiederci cosa succede se ci restringiamo all'equivalenza generata solo dalle isotopie planari e dalle mosse II e III, questo ci dà una nuova equivalenza tra nodi più forte dell'isotopia ambiente. +// #definition[ +// Dato $K$ link orientato, il *writhe* $w(K)$ (o numero di avvolgimento) è dato dalla somma dei segni dei suoi crossing -#definition[ - Due nodi o link $K_1, K_2$ si dicono equivalenti a meno di *isotopia regolare* se due loro diagrammi sono equivalenti a meno di isotopie planari e mosse II e III di Reidemeister. -] +// $ +// w(K) colon.eq sum_c epsilon(c) +// $ +// ] -#figure(image("assets/ambient-regular-isotopy-difference.jpg", width: 13cm)) +// #proposition[ +// Il writhe non dipende dall'orientazione del link +// ] -Una prima cosa che possiamo notare è che tutte le mosse R1 possono essere "fattorizzate" in modo da averle tutte raggruppate insieme. #margin-note[Magari aggiungere la dimostrazione di questa cosa] +// #proposition[ +// Il writhe è un invariante di isotopia regolare, ovvero se $K_1, K_2$ sono equivalenti a meno di isotopia regolare allora $w(K_1) = w(K_2)$. +// ] -#todo[ - Altro disegno -] +// #todo[ +// Esempi di calcolo del writhe +// ] +// Un primo fatto generale che possiamo osservare è che dato un invariante di isotopia regolare con certe proprietà, possiamo costruire un invariante di isotopia ambiente aggiungendo un fattore di correzione dato dal writhe. -C'è un'osservazione importante da fare quando parliamo di isotopia regolare. Quest'ultima è definita solo quando si parla di diagrammi di link, quando invece parliamo di isotopia ambiente ha anche senso parlare dei link come circonferenze embedded. +// #proposition[ +// Sia $R$ un anello, $a in R$ un elemento invertibile e $K$ un link orientato, allora se $L(K) in R$ è un invariante di isotopia regolare tale che -Più avanti vedremo invarianti di isotopia ambiente come il polinomio $F_K$ che però è definito attraverso l'isotopia regolare, in particolare il calcolo va fatto fissando un particolare diagramma del link. Tutt'ora è un problema aperto dare una definizione di $F_K$ che sia indipendente dalla scelta di un diagramma. +// $ +// L(#skein.over-twist) = a L(#skein.strand) +// #h(2em) +// L(#skein.under-twist) = a^(-1) L(#skein.strand) +// $ +// allora $F(K) colon.eq a^(-w(K)) L(K)$ è un invariante di isotopia ambiente. +// ] += Polinomio di Kauffman -Questo ci suggerisce un primo invariante di isotopia regolare per link orientatati, per prima cosa introduciamo il segno di un incrocio. +#todo[ + Work in progress +] -#definition[ - Dato un incrocio di un link orientato ci possono essere due casi +== Definizione assiomatica - $ - epsilon(#skein-generic(direction: (+1, +1))) = +1 - #h(2em) - epsilon(#skein-generic(direction: (+1, -1))) = -1 - $ +#todo[ + Work in progress ] -#definition[ - Dato $K$ link orientato, il *writhe* $w(K)$ (o numero di avvolgimento) è dato dalla somma dei segni dei suoi crossing +// Vediamo ora la definizione del polinomio di Kauffman. - $ - w(K) colon.eq sum_c epsilon(c) - $ -] +// #definition[ +// Dato $K$ un link non orientato, $kL_K in bb(Z)[a, a^(-1), z, z^(-1)]$, ovvero i polinomi di Laurent sugli interi nelle variabili $a$ e $z$, e verifica i seguenti assiomi: -#proposition[ - Il writhe non dipende dall'orientazione del link -] +// 1. Se $K$, $K'$ sono equivalenti a meno di isotopia regolare, allora $kL_K = kL_K'$. -#proposition[ - Il writhe è un invariante di isotopia regolare, ovvero se $K_1, K_2$ sono equivalenti a meno di isotopia regolare allora $w(K_1) = w(K_2)$. -] +// 2. Valgono le seguenti relazioni skein per quaterne di diagrammi identici ovunque tranne che nei punti indicati -#todo[ - Esempi di calcolo del writhe -] +// - $kL(#skein.over) + kL(#skein.under) = z (kL(#skein.h) + kL(#skein.v))$ -Un primo fatto generale che possiamo osservare è che dato un invariante di isotopia regolare con certe proprietà, possiamo costruire un invariante di isotopia ambiente aggiungendo un fattore di correzione dato dal writhe. +// - $kL(#skein.unit) = 1$ -#proposition[ - Sia $R$ un anello, $a in R$ un elemento invertibile e $K$ un link orientato, allora se $L(K) in R$ è un invariante di isotopia regolare tale che +// - $kL(#skein.over-twist) = a kL(#skein.strand)$ - $ - L(#skein.over-twist) = a L(#skein.strand) - #h(2em) - L(#skein.under-twist) = a^(-1) L(#skein.strand) - $ +// - $kL(#skein.under-twist) = a^(-1) kL(#skein.strand)$ +// ] - allora $F(K) colon.eq a^(-w(K)) L(K)$ è un invariante di isotopia ambiente. -] +// Come già anticipato in precedenza, vedremo che questo 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 orientati $K$ come +// $ +// F_K colon.eq a^(-w(K)) kL_K +// $ +// dove $L_K$ di un diagramma orientato è definito dimenticando l'orientazione. +// ] -= Polinomio di Kauffman +// #proposition[ +// Il polinomio $F_K (a, z)$ è un invariante di isotopia ambiente +// ] -== Definizione assiomatica +// #proof[ +// Segue dalla @prop-ext-ambient-isotopy-inv. +// ] -Vediamo ora la definizione del polinomio di Kauffman. +== Alcune proprietà del polinomio di Kauffman -#definition[ - Dato $K$ un link non orientato, $kL_K in bb(Z)[a, a^(-1), z, z^(-1)]$, ovvero i polinomi di Laurent sugli interi nelle variabili $a$ e $z$, e verifica i seguenti assiomi: +#todo[ + Work in progress +] - 1. Se $K$, $K'$ sono equivalenti a meno di isotopia regolare, allora $kL_K = kL_K'$. +// #lemma[ +// Sia $K$ un link orientato e $m(K)$ il suo mirror. Allora abbiamo la seguente relazione per $L$ +// e $F$ - 2. Valgono le seguenti relazioni skein per quaterne di diagrammi identici ovunque tranne che nei punti indicati +// $ +// L_(m(K)) (a, z) = L_K (1 slash a, z) +// #h(2em) +// F_(m(K)) (a, z) = F_K (1 slash a, z) +// $ - - $kL(#skein.over) + kL(#skein.under) = z (kL(#skein.h) + kL(#skein.v))$ +// ] - - $kL(#skein.unit) = 1$ +// #proof[ +// $m(K)$ è ottenuto da $K$ scambiando tutti i suoi crossing quindi se analizziamo cosa succede nei vari assiomi solo vicino ad un certo incrocio otteniamo le seguenti relazioni +// $ +// kL(m(#skein.unit)) = kL(#skein.unit) = 1 \ +// kL(m(#skein.over-twist)) = kL(#skein.under-twist) = a^(-1) kL(#skein.strand) \ +// kL(m(#skein.under-twist)) = kL(#skein.over-twist) = a kL(#skein.strand) +// $ - - $kL(#skein.over-twist) = a kL(#skein.strand)$ +// ed infine l'ultima relazione rimane invariata per la simmetria del polinomio di Kauffman - - $kL(#skein.under-twist) = a^(-1) kL(#skein.strand)$ -] +// $ +// kL(m(#skein.over)) + kL(m(#skein.under)) = z (kL(m(#skein.h)) + kL(m(#skein.v))) \ +// => kL(#skein.under) + kL(#skein.over) = z (kL(#skein.h) + kL(#skein.v)) \ +// => kL(#skein.over) + kL(#skein.under) = z (kL(#skein.h) + kL(#skein.v)) +// $ -Come già anticipato in precedenza, vedremo che questo è un invariante di isotopia regolare e può essere generalizzato ad invariante di isotopia ambiente come segue +// quindi passare al mirror ha l'effetto di scambiare solamente $a$ con $a^(-1)$ ma il resto della valutazione rimane identico. -#definition[ - Definiamo $F_K in Z[a, a^(-1), z, z^(-1)]$ per diagrammi orientati $K$ come - $ - F_K colon.eq a^(-w(K)) kL_K - $ - dove $L_K$ di un diagramma orientato è definito dimenticando l'orientazione. -] +// Infine per $F_m(K)$ basta osservare che $w(m(K)) = -w(K)$ per una motivazione analoga. +// ] -#proposition[ - Il polinomio $F_K (a, z)$ è un invariante di isotopia ambiente -] +// Prima di passare alla dimostrazione della buona definizione vediamo alcuni esempi di calcolo di $L_K$ in modo "implicito", questo può essere fatto scegliendo accuratamente per quali diagrammi valutare le relazioni skein degli assiomi. -#proof[ - Segue dalla @prop-ext-ambient-isotopy-inv. -] +// Ad esempio proviamo a ricavare il valore di $delta colon.eq L(#skein.unit#skein.unit)$ #margin-note[Tutte le immagini qua sotto sono temporanee e devo rifarle] -== Alcune proprietà del polinomio di Kauffman +// #let knot-picture(src, ..rest) = $thin #image("assets/derived/atlas-" + src, ..rest) thin$ -#lemma[ - Sia $K$ un link orientato e $m(K)$ il suo mirror. Allora abbiamo la seguente relazione per $L$ - e $F$ +// $ +// L( #knot-picture("infinity-0.png", height: 2.25em) ) +// + +// L( #knot-picture("infinity-1.png", height: 2.25em) ) +// &= +// z ( +// L( #knot-picture("infinity-2.png", height: 2.25em) ) +// + +// L( #knot-picture("infinity-3.png", height: 2.25em) ) +// ) \ +// => +// a L( #skein.unit ) +// + +// a^(-1) L( #skein.unit ) +// &= +// z ( +// delta +// + +// L ( #skein.unit ) +// ) \ +// => a + a^(-1) &= z ( delta + 1 ) \ +// => delta &= (a + a^(-1)) slash z - 1 +// $ - $ - L_(m(K)) (a, z) = L_K (1 slash a, z) - #h(2em) - F_(m(K)) (a, z) = F_K (1 slash a, z) - $ +// 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 un link con due componenti disgiunte. Possiamo anche trovare il risultato per il link di Hopf come segue -] +// // #figure(image("assets/implicit-calc-2.png")) -#proof[ - $m(K)$ è ottenuto da $K$ scambiando tutti i suoi crossing quindi se analizziamo cosa succede nei vari assiomi solo vicino ad un certo incrocio otteniamo le seguenti relazioni - $ - kL(m(#skein.unit)) = kL(#skein.unit) = 1 \ - kL(m(#skein.over-twist)) = kL(#skein.under-twist) = a^(-1) kL(#skein.strand) \ - kL(m(#skein.under-twist)) = kL(#skein.over-twist) = a kL(#skein.strand) - $ +// $ +// L( #knot-picture("hopf-0.png", height: 1.75em) ) +// + +// L( #knot-picture("hopf-1.png", height: 1.75em) ) +// &= +// z ( +// L( #knot-picture("hopf-2.png", height: 1.75em) ) +// + +// L( #knot-picture("hopf-3.png", height: 1.75em) ) +// ) \ +// => +// L( #knot-picture("hopf-0.png", height: 1.75em) ) +// + +// delta +// &= +// z ( +// a +// + +// a^(-1) +// ) \ +// => +// L( #knot-picture("hopf-0.png", height: 1.75em) ) +// &= +// - (a + a^(-1)) z^(-1) + 1 + (a + a^(-1)) z +// $ - ed infine l'ultima relazione rimane invariata per la simmetria del polinomio di Kauffman - $ - kL(m(#skein.over)) + kL(m(#skein.under)) = z (kL(m(#skein.h)) + kL(m(#skein.v))) \ - => kL(#skein.under) + kL(#skein.over) = z (kL(#skein.h) + kL(#skein.v)) \ - => kL(#skein.over) + kL(#skein.under) = z (kL(#skein.h) + kL(#skein.v)) - $ +// Ed del nodo trifoglio - quindi passare al mirror ha l'effetto di scambiare solamente $a$ con $a^(-1)$ ma il resto della valutazione rimane identico. +// // #figure(image("assets/implicit-calc-3.png")) - Infine per $F_m(K)$ basta osservare che $w(m(K)) = -w(K)$ per una motivazione analoga. -] +// $ +// L( #knot-picture("trefoil-0.png", height: 2em) ) +// + +// L( #knot-picture("trefoil-1.png", height: 2em) ) +// &= +// z ( +// L( #knot-picture("trefoil-2.png", height: 2em) ) +// + +// L( #knot-picture("trefoil-3.png", height: 2em) ) +// ) \ +// => +// L( #knot-picture("trefoil-0.png", height: 2em) ) +// + +// a +// &= +// z ( +// L( #knot-picture("hopf-0.png", height: 1.75em) ) +// + +// a^(-2) +// ) \ +// => +// L( #knot-picture("trefoil-0.png", height: 2em) ) +// &= +// -(2a + a^(-1)) + (1 + a^(-2)) z + (a + a^(-1)) z^2 +// $ -Prima di passare alla dimostrazione della buona definizione vediamo alcuni esempi di calcolo di $L_K$ in modo "implicito", questo può essere fatto scegliendo accuratamente per quali diagrammi valutare le relazioni skein degli assiomi. - -Ad esempio proviamo a ricavare il valore di $delta colon.eq L(#skein.unit#skein.unit)$ #margin-note[Tutte le immagini qua sotto sono temporanee e devo rifarle] - -#let knot-picture(src, ..rest) = $thin #image("assets/derived/atlas-" + src, ..rest) thin$ - -$ - L( #knot-picture("infinity-0.png", height: 2.25em) ) - + - L( #knot-picture("infinity-1.png", height: 2.25em) ) - &= - z ( - L( #knot-picture("infinity-2.png", height: 2.25em) ) - + - L( #knot-picture("infinity-3.png", height: 2.25em) ) - ) \ - => - a L( #skein.unit ) - + - a^(-1) L( #skein.unit ) - &= - z ( - delta - + - L ( #skein.unit ) - ) \ - => a + a^(-1) &= z ( delta + 1 ) \ - => delta &= (a + a^(-1)) slash z - 1 -$ - -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 un link con due componenti disgiunte. Possiamo anche trovare il risultato per il link di Hopf come segue - -// #figure(image("assets/implicit-calc-2.png")) - -$ - L( #knot-picture("hopf-0.png", height: 1.75em) ) - + - L( #knot-picture("hopf-1.png", height: 1.75em) ) - &= - z ( - L( #knot-picture("hopf-2.png", height: 1.75em) ) - + - L( #knot-picture("hopf-3.png", height: 1.75em) ) - ) \ - => - L( #knot-picture("hopf-0.png", height: 1.75em) ) - + - delta - &= - z ( - a - + - a^(-1) - ) \ - => - L( #knot-picture("hopf-0.png", height: 1.75em) ) - &= - - (a + a^(-1)) z^(-1) + 1 + (a + a^(-1)) z -$ - - -Ed del nodo trifoglio - -// #figure(image("assets/implicit-calc-3.png")) - -$ - L( #knot-picture("trefoil-0.png", height: 2em) ) - + - L( #knot-picture("trefoil-1.png", height: 2em) ) - &= - z ( - L( #knot-picture("trefoil-2.png", height: 2em) ) - + - L( #knot-picture("trefoil-3.png", height: 2em) ) - ) \ - => - L( #knot-picture("trefoil-0.png", height: 2em) ) - + - a - &= - z ( - L( #knot-picture("hopf-0.png", height: 1.75em) ) - + - a^(-2) - ) \ - => - L( #knot-picture("trefoil-0.png", height: 2em) ) - &= - -(2a + a^(-1)) + (1 + a^(-2)) z + (a + a^(-1)) z^2 -$ +// #definition[ +// Diremo che un diagramma è _split_ se è l'unione disgiunta di due diagrammi separati, in questo caso scriviamo $K = K_1 union.sq K_2$ dove $K_1$ e $K_2$ sono le due componenti disgiunte. +// ] -#definition[ - Diremo che un diagramma è _split_ se è l'unione disgiunta di due diagrammi separati, in questo caso scriviamo $K = K_1 union.sq K_2$ dove $K_1$ e $K_2$ sono le due componenti disgiunte. -] +// #definition[ +// Un diagramma $K$ è una *somma connessa* se appare come due diagrammi disgiunti connessi da due archi paralleli (a meno di isotopie planari). 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 $K = K_1 hash K_2$. +// ] -#definition[ - Un diagramma $K$ è una *somma connessa* se appare come due diagrammi disgiunti connessi da due archi paralleli (a meno di isotopie planari). 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 $K = K_1 hash K_2$. -] +// #proposition[ +// Valgono le seguenti relazioni di $L$ ed $F$ per diagrammi split o in somma connessa + +// #{ +// set align(center) +// grid( +// rows: 2, +// columns: 2, +// column-gutter: 2em, +// row-gutter: 1em, +// $L(K_1 union.sq K_2) = delta L(K_1) L(K_2)$, $F(K_1 union.sq K_2) = delta F(K_1) F(K_2)$, +// $L(K_1 hash K_2) = L(K_1) L(K_2)$, $F(K_1 hash K_2) = F(K_1) F(K_2)$, +// ) +// } + +// dove $delta = (a + a^(-1)) slash z - 1$ ed è lo stesso coefficiente che abbiamo trovato nell'esempio precedente. +// ] -#proposition[ - Valgono le seguenti relazioni di $L$ ed $F$ per diagrammi split o in somma connessa - - #{ - set align(center) - grid( - rows: 2, - columns: 2, - column-gutter: 2em, - row-gutter: 1em, - $L(K_1 union.sq K_2) = delta L(K_1) L(K_2)$, $F(K_1 union.sq K_2) = delta F(K_1) F(K_2)$, - $L(K_1 hash K_2) = L(K_1) L(K_2)$, $F(K_1 hash K_2) = F(K_1) F(K_2)$, - ) - } - - dove $delta = (a + a^(-1)) slash z - 1$ ed è lo stesso coefficiente che abbiamo trovato nell'esempio precedente. +== Dimostrazione Forma Induttiva + +#todo[ + Work in progress ] -== Dimostrazione Forma Induttiva +// Prima di passare a descrivere la definizione induttiva diamo alcune definizioni di manipolazioni di un diagramma. -Prima di passare a descrivere la definizione induttiva diamo alcune definizioni di manipolazioni di un diagramma. +// #definition[ +// Sia $K$ un diagramma, $cal(U)$ la sua ombra planare#footnote[Ovvero per $K subset bb(R)^3$ poniamo $U colon.eq pi(K) subset bb(R)^2$] e $p in cal(U)$ un punto di partenza direzionato in $cal(U)$. +// ] -#definition[ - Sia $K$ un diagramma, $cal(U)$ la sua ombra planare#footnote[Ovvero per $K subset bb(R)^3$ poniamo $U colon.eq pi(K) subset bb(R)^2$] e $p in cal(U)$ un punto di partenza direzionato in $cal(U)$. -] +// - Il suo *nodo (o link) banale standard* associato detto $hat(K)(cal(U), p)$ è definito come segue: si prende il primo 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 (o link) banale standard* associato detto $hat(K)(cal(U), p)$ è definito come segue: si prende il primo 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. +// #figure( +// image( +// "assets/standard-unlink-construction.png", +// width: 12cm, +// ), +// caption: [ +// Un link con $3$ componenti e punti di partenza direzionati $p_1, p_2, p_3$. In #box(radius: 3pt, outset: (y: 3pt), inset: (x: 2pt), fill: rgb("#91cdff"))[azzurro] sono evidenziati gli incroci che sono cambiati rispetto al link di partenza. +// ], +// ) - #figure( - image( - "assets/standard-unlink-construction.png", - width: 12cm, - ), - caption: [ - Un link con $3$ componenti e punti di partenza direzionati $p_1, p_2, p_3$. In #box(radius: 3pt, outset: (y: 3pt), inset: (x: 2pt), fill: rgb("#91cdff"))[azzurro] sono evidenziati gli incroci che sono cambiati rispetto al link di partenza. - ], - ) +// - Sia $K$ un link, $lambda = (lambda_n, dots, lambda_0)$ una sequenza di etichette di incroci allora poniamo -- Sia $K$ un link, $lambda = (lambda_n, dots, lambda_0)$ una sequenza di etichette di incroci allora poniamo +// $ +// A_i^lambda = E_lambda_i S_lambda_(i-1) dots.c space S_lambda_0 +// #h(2em) +// B_i^lambda = e_lambda_i S_lambda_(i-1) dots.c space S_lambda_0 +// $ - $ - A_i^lambda = E_lambda_i S_lambda_(i-1) dots.c space S_lambda_0 - #h(2em) - B_i^lambda = e_lambda_i S_lambda_(i-1) dots.c space S_lambda_0 - $ +// notiamo che per $i=0$ abbiamo che $A_0^lambda K = E_lambda_0 K$ e $B_0^lambda K = e_lambda_0 K$ che è l'unico caso degenere per questa definizione. - notiamo che per $i=0$ abbiamo che $A_0^lambda K = E_lambda_0 K$ e $B_0^lambda K = e_lambda_0 K$ che è l'unico caso degenere per questa definizione. +// - #todo[Operazione compatta $sum_K (lambda)$] -- #todo[Operazione compatta $sum_K (lambda)$] +// L'idea per la definizione induttiva è di considerare dato un nodo o un link $K$ il suo nodo o link discendente standard $hat(K)$. Questo induce una sequenza di indici $lambda$ che indichiamo con $0, dots, n$. Se la applichiamo incrementalmente a $K$ otteniamo le seguenti relazioni -L'idea per la definizione induttiva è di considerare dato un nodo o un link $K$ il suo nodo o link discendente standard $hat(K)$. Questo induce una sequenza di indici $lambda$ che indichiamo con $0, dots, n$. Se la applichiamo incrementalmente a $K$ otteniamo le seguenti relazioni +// $ +// L_K + L_(S_0 K) = z( L_(E_0 K) + L_(e_0 K) ) \ +// L_(S_0 K) + L_(S_1 S_0 K) = z( L_(E_0 S_0 K) + L_(e_0 S_0 K) ) \ +// dots.v \ +// L_(S_(n-1) dots S_0 K) + L_(hat(K)) = z (L_(E_n S_(n-1) dots S_0 K) + L_(e_n S_(n-1) dots S_0 K)) +// $ -$ - L_K + L_(S_0 K) = z( L_(E_0 K) + L_(e_0 K) ) \ - L_(S_0 K) + L_(S_1 S_0 K) = z( L_(E_0 S_0 K) + L_(e_0 S_0 K) ) \ - dots.v \ - L_(S_(n-1) dots S_0 K) + L_(hat(K)) = z (L_(E_n S_(n-1) dots S_0 K) + L_(e_n S_(n-1) dots S_0 K)) -$ +// se ora sommiamo e sottraiamo membro a membro otteniamo la seguente identità -se ora sommiamo e sottraiamo membro a membro otteniamo la seguente identità +// #context block( +// width: page.width, +// grid( +// columns: 3, +// column-gutter: 2em, +// row-gutter: 1em, +// $L_K + L_(S_0 K)$, [], $z( L_(E_0 K) + L_(e_0 K) )$, +// $-(L_(S_0 K) + L_(S_1 S_0 K))$, [], $-z( L_(E_0 S_0 K) + L_(e_0 S_0 K) )$, +// $dots.v$, $=$, $dots.v$, +// $+(-1)^n (L_(S_(n-1) dots S_0 K) + L_(hat(K)))$, +// [], +// $+(-1)^n (z (L_(E_n S_(n-1) dots S_0 K) + L_(e_n S_(n-1) dots S_0 K)))$, +// ), +// ) -#context block( - width: page.width, - grid( - columns: 3, - column-gutter: 2em, - row-gutter: 1em, - $L_K + L_(S_0 K)$, [], $z( L_(E_0 K) + L_(e_0 K) )$, - $-(L_(S_0 K) + L_(S_1 S_0 K))$, [], $-z( L_(E_0 S_0 K) + L_(e_0 S_0 K) )$, - $dots.v$, $=$, $dots.v$, - $+(-1)^n (L_(S_(n-1) dots S_0 K) + L_(hat(K)))$, - [], - $+(-1)^n (z (L_(E_n S_(n-1) dots S_0 K) + L_(e_n S_(n-1) dots S_0 K)))$, - ), -) +// notiamo che possiamo cancellare tutti i termini del membro di sinistra che compaiono due volte con segno opposto -notiamo che possiamo cancellare tutti i termini del membro di sinistra che compaiono due volte con segno opposto +// #context block( +// width: page.width, +// grid( +// columns: 3, +// column-gutter: 2em, +// row-gutter: 1em, +// $L_K + cancel(L_(S_0 K))$, [], $z( L_(E_0 K) + L_(e_0 K) )$, +// $-(cancel(L_(S_0 K)) + cancel(L_(S_1 S_0 K)))$, [], $-z( L_(E_0 S_0 K) + L_(e_0 S_0 K) )$, +// $dots.v$, $=$, $dots.v$, +// $+(-1)^n (cancel(L_(S_(n-1) dots S_0 K)) + L_(hat(K)))$, +// [], +// $+(-1)^n (z (L_(E_n S_(n-1) dots S_0 K) + L_(e_n S_(n-1) dots S_0 K)))$, +// ), +// ) -#context block( - width: page.width, - grid( - columns: 3, - column-gutter: 2em, - row-gutter: 1em, - $L_K + cancel(L_(S_0 K))$, [], $z( L_(E_0 K) + L_(e_0 K) )$, - $-(cancel(L_(S_0 K)) + cancel(L_(S_1 S_0 K)))$, [], $-z( L_(E_0 S_0 K) + L_(e_0 S_0 K) )$, - $dots.v$, $=$, $dots.v$, - $+(-1)^n (cancel(L_(S_(n-1) dots S_0 K)) + L_(hat(K)))$, - [], - $+(-1)^n (z (L_(E_n S_(n-1) dots S_0 K) + L_(e_n S_(n-1) dots S_0 K)))$, - ), -) +// Da cui otteniamo un'espressione per $L_K$, inoltre posto $lambda = (0, dots, n)$ -Da cui otteniamo un'espressione per $L_K$, inoltre posto $lambda = (0, dots, n)$ - -#align( - center, - block( - width: 15cm, - [ - #set align(center) - - $ - => - L_K - &= - (-1)^(n+1) L_(hat(K)) + z sum_(i=0)^n (-1)^i ( - L_(E_n S_(n-1) dots S_0 K) + L_(e_n S_(n-1) dots S_0 K) - ) \ - &= - (-1)^(n+1) L_(hat(K)) + z sum_(i=0)^n (-1)^i ( - L_(A_n^lambda K) + L_(B_n^lambda K) - ) - $ - - ], - ), -) +// #align( +// center, +// block( +// width: 15cm, +// [ +// #set align(center) + +// $ +// => +// L_K +// &= +// (-1)^(n+1) L_(hat(K)) + z sum_(i=0)^n (-1)^i ( +// L_(E_n S_(n-1) dots S_0 K) + L_(e_n S_(n-1) dots S_0 K) +// ) \ +// &= +// (-1)^(n+1) L_(hat(K)) + z sum_(i=0)^n (-1)^i ( +// L_(A_n^lambda K) + L_(B_n^lambda K) +// ) +// $ + +// ], +// ), +// ) -Questo ci dà un'idea su come sia possibile calcolare induttivamente $L_K$ in termini di $L_hat(K)$ e di altri diagrammi con meno incroci del diagramma di partenza. +// Questo ci dà un'idea su come sia possibile calcolare induttivamente $L_K$ in termini di $L_hat(K)$ e di altri diagrammi con meno incroci del diagramma di partenza. -Affinché sia sempre possibile raggiungere un nodo in forma di nodo banale standard serve aggiungere la seguente relazione quando $K = K_1 union.sq K_2$ con $K_1$ sovrastante $K_2$. +// Affinché sia sempre possibile raggiungere un nodo in forma di nodo banale standard serve aggiungere la seguente relazione quando $K = K_1 union.sq K_2$ con $K_1$ sovrastante $K_2$. -$ - L_(K_1 union.sq K_2) = delta L_(K_1) L_(K_2) -$ +// $ +// L_(K_1 union.sq K_2) = delta L_(K_1) L_(K_2) +// $ -sempre con $delta = (a + a^(-1)) slash z - 1$. +// sempre con $delta = (a + a^(-1)) slash z - 1$. -// Closed form algorithm +// // Closed form algorithm -#definition(name: [Forma chiusa per $kL_K$])[ - Il polinomio $kL_(K)(a,z)$ è definito induttivamente come segue, abbiamo i seguenti casi: -] +// #definition(name: [Forma chiusa per $kL_K$])[ +// Il polinomio $kL_(K)(a,z)$ è definito induttivamente come segue, abbiamo i seguenti casi: +// ] -// la definizione è sotto perché gli enumerate non vanno d'accordo con le figure +// // la definizione è sotto perché gli enumerate non vanno d'accordo con le figure -1. Se $K = hat(K)$ è un _nodo banale standard_ allora $kL_K (a, z) colon.eq a^w(K)$ +// 1. Se $K = hat(K)$ è un _nodo banale standard_ allora $kL_K (a, z) colon.eq a^w(K)$ -2. Se $K_1$ è _sovrastante_ $K_2$, sia $d colon.eq (a + a^(-1)) slash z - 1$ e allora +// 2. Se $K_1$ è _sovrastante_ $K_2$, sia $d colon.eq (a + a^(-1)) slash z - 1$ e allora - $ - kL(K_1 union.sq K_2) colon.eq d kL(K_1) kL(K_2) - $ +// $ +// kL(K_1 union.sq K_2) colon.eq d kL(K_1) kL(K_2) +// $ -3. Se $K = K_1 union.sq dotss union.sq K_n$ +// 3. Se $K = K_1 union.sq dotss union.sq K_n$ - - Se un $K_i$ è _sovrastante_ un'altra componente allora applica (ii). +// - Se un $K_i$ è _sovrastante_ un'altra componente allora applica (ii). - - Se nessun $K_i$ è _sovrastante_ tutti gli altri, siano $p_1, ..., p_n$ dei punti di partenza direzionati su $K_1, ..., K_n$ e sia $overline(p)_i$ lo stesso punto di partenza direzionato con la direzione opposta di $p_i$ su $K_i$. Sia $lambda(p_i)$ la sequenza di scambi di incroci di $K_i$ con $K - K_i$ tale che $hat(K)(lambda(p_i)) = K_i union.sq (K - K_i)$ tale che $K_i$ sia _sovrastante_ il resto delle componenti. A questo punto possiamo definire $kL_K$ come +// - Se nessun $K_i$ è _sovrastante_ tutti gli altri, siano $p_1, ..., p_n$ dei punti di partenza direzionati su $K_1, ..., K_n$ e sia $overline(p)_i$ lo stesso punto di partenza direzionato con la direzione opposta di $p_i$ su $K_i$. Sia $lambda(p_i)$ la sequenza di scambi di incroci di $K_i$ con $K - K_i$ tale che $hat(K)(lambda(p_i)) = K_i union.sq (K - K_i)$ tale che $K_i$ sia _sovrastante_ il resto delle componenti. A questo punto possiamo definire $kL_K$ come - #[ - #set text(size: 11pt) - $ - kL_K (a, z) colon.eq - 1 / (2n) [ - sum_(i=1)^n sum_(q=p_i, overline(p)_i) ((-1)^(|lambda(q)|+1) d kL_(K_i) kL_(K - K_i) + z sum_K (lambda(q))) - ] - $ - ] +// #[ +// #set text(size: 11pt) +// $ +// kL_K (a, z) colon.eq +// 1 / (2n) [ +// sum_(i=1)^n sum_(q=p_i, overline(p)_i) ((-1)^(|lambda(q)|+1) d kL_(K_i) kL_(K - K_i) + z sum_K (lambda(q))) +// ] +// $ +// ] - - Se $K$ è una singola componente allora sia $p$ un punto di partenza direzionato su $K$ e $overline(p)$ quello con direzione opposta. Sia $lambda(p)$ la sequenza di scambi di incroci che lo porta al nodo banale standard $hat(K)$ e definiamo +// - Se $K$ è una singola componente allora sia $p$ un punto di partenza direzionato su $K$ e $overline(p)$ quello con direzione opposta. Sia $lambda(p)$ la sequenza di scambi di incroci che lo porta al nodo banale standard $hat(K)$ e definiamo - #[ - #set text(size: 11pt) +// #[ +// #set text(size: 11pt) - $ - kL_K (a, z) colon.eq - 1 / 2 [ - sum_(q = p, overline(p)) ((-1)^(|lambda(q)|+1) kL(hat(K)(lambda(q))) + z sum_K (lambda(q))) - ] - $ - ] +// $ +// kL_K (a, z) colon.eq +// 1 / 2 [ +// sum_(q = p, overline(p)) ((-1)^(|lambda(q)|+1) kL(hat(K)(lambda(q))) + z sum_K (lambda(q))) +// ] +// $ +// ] diff --git a/src/prelude.typ b/src/prelude.typ index 695a026..ee362de 100644 --- a/src/prelude.typ +++ b/src/prelude.typ @@ -30,7 +30,13 @@ figure( body, kind: "definition", - supplement: [Definizione], + supplement: { + [Definizione] + if name != none { + [ -- ] + name + } + }, ) } @@ -49,7 +55,7 @@ ) } -#let proposition(body, name: none, numbered: true) = { +#let proposition(body, numbered: true) = { // show figure: statement-style(name, numbered: numbered) return figure( @@ -61,7 +67,7 @@ ) } -#let lemma(body, name: none, numbered: true) = { +#let lemma(body, numbered: true) = { // show figure: statement-style(name, numbered: numbered) figure( body, @@ -77,7 +83,12 @@ figure( body, kind: "theorem", - supplement: [Teorema], + supplement: { + [Teorema] + if name != none { + [ (#name)] + } + }, numbering: if numbered { "1" }, ) } diff --git a/src/theme.typ b/src/theme.typ index 8e49487..5804ea3 100644 --- a/src/theme.typ +++ b/src/theme.typ @@ -74,10 +74,7 @@ let i = counter(page).get().first() align( center, - { - set text(size: large-size) - [#i] - }, + [#i], ) }, )