From 91834e71649a0c7be3f4d0ae25276fa21efa1484 Mon Sep 17 00:00:00 2001 From: Antonio De Lucreziis Date: Tue, 8 Jul 2025 20:08:52 +0200 Subject: [PATCH] categorical proof with diagrams and crossings --- out/tesi-triennale.pdf | Bin 2008694 -> 1759396 bytes src/main.typ | 93 ++++++++++++++++++++++++++++------------- src/prelude.typ | 4 +- src/theme.typ | 11 ++++- 4 files changed, 75 insertions(+), 33 deletions(-) diff --git a/out/tesi-triennale.pdf b/out/tesi-triennale.pdf index c498b3a6be636634c8f945ebe6f3087d7334cba1..3f67d8c8f8e0644fb2443f426e3a141952fee43f 100644 GIT binary patch delta 159507 zcmYJ41yCDp)bEQ!arffx?(XjHP~6>FT#A-ph2l^s?(P(q7I$}dw+rw6zI%7hKY4QY z*t0X6%q02Uj`|c-+aggYi%T-FFmipS2z)=jcT$6xP0^ucbY(Op)xImqgRy~v{-VRN za}YBVt9}*~1bK6lf;#`=fRuQV|0fCr0~Q3v4hk~EuFU}p00-E3n2EVqKa88@eZ}@+ z96apAT>lv-^Sh6On2VE`jg5tvi|fPK*tv-$!C zxIXM-P9E+LV|~wlFa2(8?99Y0?$P3oH9eJ6pL5peGew9;q50JehGH&b_Gzv-S& z9*=U{TRK+R`wsSxHZtxCnf}(kSk8DTOgxE{-(~n4Am+LmVgD_f%(i{is<~9sc+q8c 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zr-83#`8f0-V!rc4U}T4?;R?LZ7$&PXIKrkZx&so^zgPVxq6)_h`XYt$olPWAI}9bV z$ZLTuJJ@|7_N%3VvRP+K_*xjFk8gURqU8}qg|7pEr-)q6XE2!$J}9s8R=bagn+A(q zocd8f;BJEh2;*_;M;$H;JWR$5^SriXRLERdHW9f@&U6R>I;+i*|NK;g5`s%6A}oT~ z%1MIK%%Lq6xrmooRRyjzV<}d#IW_2urfSNLA|7#p-GyU6uk8>Q9O_40aQFgzkdKZg z3J}NQK^GN^ON*kZ3w#L;7;fP4TF@PpG4Kq7D^M8>v6U0VV77C^y!9X6gTMmJ&od!7 zgP}a&J4-sBbbyD$S$I6dbVzI%s6qVkEJlPydVe5brYy6Yv^MoahmPeE7EpYC#~KuC zV!kgb72XL=4f^LL#?*FjZ6fwK@C=PFHn4KmR6Z_XMJC%dC}CydT_D!hf9#F~0yEDw z1KMEdEVyrW`e}y2P#rk%6s|&IczoWl7!Z@&(b>SEJB&uH_X7{NnVm3+tNR%qHqbcw zvV(p2@sRU}JAhCk55qFf(HYAFyFEb|t1f{yxkTEulGvTyf26151|*w_o;&|(Dd{e( r>S+H #figure(image("assets/move-1-factorization.png", width: 10cm)) @@ -663,9 +665,9 @@ Sia $lambda = (lambda_n, dots, lambda_0)$ una sequenza di etichette di incroci d dove $abs(lambda) colon.eq n$. -#fact[ +#fact(numbered: true)[ Un nodo banale standard è equivalente ad un nodo banale a meno di isotopia ambiente. -] +] *Osservazione.* Se consideriamo un diagramma di link $K$, $hat(K)(cal(U), p)$ ci induce una sequenza di scambi $lambda$ su $K$ tale che $hat(K)(cal(U), p) = S_lambda_n dots.c space S_lambda_0 K$. @@ -686,7 +688,33 @@ Enunciamo ora un lemma che utilizzeremo più volte in seguito. ] #proof[ - #todo[da rifare, bla bla diagramma $D_1 -> dots -> D_n$ bla bla mosse $R_i$ di tipo I, II, III ... $D_1' -> dots -> D_n'$] + 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 + + $ + 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. + + Applichiamo induttivamente le seguenti modifiche per ogni mossa $m_i$ nella sequenza: + + - 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 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 + + #todo[disegnino] + + - 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. + + #todo[disegnino] + + Otteniamo così una nuova successione di diagrammi $D_i^*$ e di mosse solo di tipo II o III + + $ + 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. ] 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. @@ -932,10 +960,7 @@ Nel corso della dimostrazione tutti gli argomenti per induzione si baseranno sul + (-1)^(n+1) ( & L[E_0 S_n dotss S_1 K] + L[e_0 S_n dotss S_1 K] ) $ - Permutando i termini di tutte le righe centrali possiamo - // riscriverli in modo da averli nella forma - ottenere una somma di termini della forma - $L[blank K] + L[blank S_0 K]$: + Permutando i termini di tutte le righe centrali possiamo ottenere una somma di termini della forma $L[blank K] + L[blank S_0 K]$: $ sum_K (lambda) - sum_K (mu) = @@ -947,7 +972,7 @@ Nel corso della dimostrazione tutti gli argomenti per induzione si baseranno sul + (-1)^(n+1) ( & L[E_0 S_n dotss S_1 K] + L[e_0 S_n dotss S_1 K] ) $ - Ora ricordiamo l'@observation-seq-ops-commute per cui operazioni su indici diversi commutano tra loro, quindi possiamo spostare l'operazione $S_0$ in modo che sia applicata per ultima ed otteniamo: + Ora ricordiamo l'@observation-seq-ops-commute, per cui operazioni su indici diversi commutano tra loro, quindi possiamo spostare l'operazione $S_0$ in modo che sia applicata per ultima ed otteniamo: $ sum_K (lambda) - sum_K (mu) = @@ -959,7 +984,7 @@ Nel corso della dimostrazione tutti gli argomenti per induzione si baseranno sul + (-1)^(n+1) ( & L[E_0 S_n dotss S_1 K] + L[e_0 S_n dotss S_1 K] ) $ - A questo punto notiamo che tutti i diagrammi $E_i S_(i-1) dotss S_1 K$ e $e_i S_(i-1) dotss S_1 K$ per $i = 1, dots, n$ hanno applicato uno splice quindi hanno strettamente meno incroci di $K$. Possiamo quindi applicare l'ipotesi induttiva e sostituire tutti i termini nella forma: + A questo punto notiamo che tutti i diagrammi $E_i S_(i-1) dotss S_1 K$ e $e_i S_(i-1) dotss S_1 K$ per $i = 1, dots, n$ hanno applicato uno splice, quindi hanno strettamente meno incroci di $K$. Possiamo dunque applicare l'ipotesi induttiva e sostituire tutti i termini nella forma: $ L[blank] + L[S_0 blank] = z (L [E_0 blank] + L[e_0 blank]) @@ -1043,7 +1068,7 @@ Nel corso della dimostrazione tutti gli argomenti per induzione si baseranno sul le espressioni che dipendono dai punti base sono $(-1)^(abs(lambda(q)) + 1)$ e $sum_K (lambda(q))$. Se applichiamo una $1$-rotazione alle sequenze di rialzo $lambda(q)$ le lunghezze non cambiano ed usando il @lemma-sum-switches-rotation anche le somme $sum_K (lambda(q)))$ sono invarianti per $1$-rotazioni. - A questo punto possiamo estendere l'invarianza per $1$-rotazioni a tutte le possibili permutazioni cicliche e questo ci dimostra che questo caso non dipende dalla scelta del punto base. + Chiaramente l'invarianza si estende a tutte le permutazioni cicliche e questo dimostra che il caso @kauffman-rec-multi-component per più componenti della @kauffman-rec-inductive-def[Definizione] non dipende dalla scelta del punto base. ] #align(center, line(stroke: 0.5pt, length: 50%)) @@ -1058,16 +1083,24 @@ $ ] $ -La dimostrazione procede sempre mostrando l'indipendenza dalla scelta di punto base. Vediamo alcuni fatti: +La dimostrazione procede sempre mostrando l'indipendenza dalla scelta di punto base. Vediamo prima alcune proprietà dei nodi banali standard: -#grid( - columns: (1fr, auto), - gutter: 1em, - fact[ - 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). - ], - image("assets/splice-circuits.png", width: 4cm), -) +#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). +] + +#lemma[ + La proprietà del lemma @std-unknot-to-curls si estende al diagramma $e_i hat(K)$. +] + +// #grid( +// columns: (1fr, auto), +// gutter: 1em, +// fact[ +// 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). +// ], +// image("assets/splice-circuits.png", width: 4cm), +// ) // #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 standard). @@ -1077,24 +1110,24 @@ La dimostrazione procede sempre mostrando l'indipendenza dalla scelta di punto b // Nel caso di una sola componente non abbiamo più un nodo in forma discendente, ma questa è formata da una prima parte discendente, seguita da una ascendente (la parte in mezzo è percorsa al contrario) ed infine un'altra discendente, pensando in termini di altezze rispetto al piano di proiezione, ogni parte può essere semplificata indipendentemente dalle altre quindi abbiamo ancora un nodo banale a meno di isotopia ambiente. -#fact[ - La proprietà del lemma @std-unknot-to-curls si estende al diagramma $e_i hat(K)$. -] +// #fact[ +// La proprietà del lemma @std-unknot-to-curls si estende al diagramma $e_i hat(K)$. +// ] -La dimostrazione di questi fatti è omessa in quanto molto tecnica, ma per dare un'idea di ciò che avviene in questi tre casi possiamo considerare la seguente figura. +// La dimostrazione di questi fatti è omessa in quanto molto tecnica, ma per dare un'idea di ciò che avviene in questi tre casi possiamo considerare la seguente figura. -#figure( - image("assets/std-unknot-splices.png", width: 12cm), - caption: [Diagramma di $K$, $E_i K$ e $e_i K$ con rispettivo grafico delle "altezze" dei fili.], -) +// #figure( +// image("assets/std-unknot-splices.png", width: 12cm), +// caption: [Diagramma di $K$, $E_i K$ e $e_i K$ con rispettivo grafico delle "altezze" dei fili.], +// ) -Nel primo caso il diagramma $K$ nodo è in forma discendente, possiamo visualizzarlo al variare "dell'altezza" come sotto. Dopo gli splice in un caso otteniamo due componenti disgiunte ognuna ancora in forma discendente, mentre nell'altro caso anche se abbiamo una parte rovesciata complessivamente il nodo rimane banale. +// Nel primo caso il diagramma $K$ nodo è in forma discendente, possiamo visualizzarlo al variare "dell'altezza" come sotto. Dopo gli splice in un caso otteniamo due componenti disgiunte ognuna ancora in forma discendente, mentre nell'altro caso anche se abbiamo una parte rovesciata complessivamente il nodo rimane banale. // #proof[ // #todo[work in progress] // ] -Vediamo ora alcune identità per nodi banali standard che ci mostrano cosa succede quando facciamo avanzare il punto base di un incrocio. +// Utilizzando questi ultimi risultati identità per nodi banali standard che ci mostrano cosa succede quando facciamo avanzare il punto base di un incrocio. #lemma[ Sia $p$ un punto di partenza direzionato e $hat(K) colon.eq hat(K)(cal(U), p)$ un nodo banale standard. Sia $i$ il primo incrocio in $K$ subito dopo il punto di partenza $p$. Sia $q$ un altro punto base direzionato posto nell'arco subito dopo l'incrocio $i$ con la sua stessa direzione e poniamo $hat(K)' colon.eq hat(K)(cal(U), q)$. Allora valgono le seguenti proprietà diff --git a/src/prelude.typ b/src/prelude.typ index 7438b1a..2947423 100644 --- a/src/prelude.typ +++ b/src/prelude.typ @@ -36,7 +36,7 @@ }) } -#let fact(body, name: none) = { +#let fact(body, numbered: false, name: none) = { // show figure: statement-style(name, numbered: false) figure(body, kind: "fact", supplement: { [Fatto] @@ -44,7 +44,7 @@ [ -- ] name } - }) + }, numbering: if numbered { "1" }) } #let observation(body) = { diff --git a/src/theme.typ b/src/theme.typ index e854fc0..14f6de2 100644 --- a/src/theme.typ +++ b/src/theme.typ @@ -183,7 +183,7 @@ show figure.caption: caption => block(inset: (x: 1.4em), { set align(left) set text(size: 10.5pt) - smallcaps(caption.supplement) + smallcaps([Figura]) if caption.numbering != none { [ ] numbering(caption.numbering, ..caption.counter.at(it.location())) @@ -236,6 +236,15 @@ block({ strong({ it.supplement + if it.numbering != none { + [ ] + // current chapter and section + + [#( + ..counter(heading).get().slice(0, 1).map(it => str(it)), + numbering(it.numbering, ..it.counter.at(it.location())), + ).join(".")] + } [.] }) it.body