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z4rD<}oajMVi|gS1-_0)yK{L9UC4Pz4Z_|4CMu4~p7(NWNWK#@eF!hnj7#T7;Ni;tJ zlUGzRLBRwXxeNh=4XG?VW`sshfr4HOUCyD20jj`*l64G;jGV-Ew8UEkz>?%W5IG`l z{u0Nq8E^^p3{WftgT!@cn;^>gkN-EJD|#RWyaAN~Cq-Qc++UJt01g5&LWIje$0^y&5D2S>b-Jpd~NbnFQm$GvZ;VW!HZAF!E7}7exWjHAy zQXDWP$zGJ#hZ?p@E+7u*OQ4NGI(GP62t{7H z4wroB0UFJdvVDxW867K%GQczB#UT3+CDeZ?0S8F-(wz_a-p7banEz11{D%_uKa>Dd zl_)`HU7`+{SjnyvM4?m(fE=G8D^8f$< diff --git a/src/presentation.typ b/src/presentation.typ index 0745c88..5b26288 100644 --- a/src/presentation.typ +++ b/src/presentation.typ @@ -44,7 +44,7 @@ == Nodi e Diagrammi -*Nodo.* $K subset bb(R)^3$ è un *nodo (tame)* se $exists f : bb(S)^1 arrow.hook bb(R)^3$ _embedding loc. piatto_ con $K = f(bb(S)^1)$. +*Nodo.* $K subset bb(R)^3$ è un *nodo (tame)* se $exists f : bb(S)^1 arrow.hook bb(R)^3$ _embedding loc. piatto_ con $K = f(bb(S)^1)$ *Link.* $bb(S)^1 arrow bb(R)^3 ~> bb(S)^1 union.sq dots.c union.sq bb(S)^1 arrow bb(R)^3$ @@ -66,15 +66,19 @@ e posta $H_t (x) colon.eq H(x, t)$ si ha: \ -*Def.* Un *diagramma* $D subset bb(R)^2$ proviene da un link $L subset RR^3$ attraverso una proiezione regolare. I punti doppi del diagramma sono detti *incroci*. +*Def.* Dato un _link_ $L subset RR^3$ esiste un suo *diagramma* $D subset RR^2$ attraverso una proiezione regolare. I punti doppi del diagramma sono detti *incroci*. + +// SPEAKER NOTE: In particolare in corrispondenza dei punti doppi, che chiameremo incroci, aggiungiamo l'informazione sopra-sotto. == Teorema di Reidemeister -*Mosse di Reidemeister.* Le mosse I, II, III in figura sono dette. +Le mosse *I, II, III* in figura, sono dette *mosse di Reidemeister*. *Teorema (di Reidemeister).* Due diagrammi di link _equivalenti_ sono collegati da una _successione finita di mosse di Reidemeister_ e isotopie planari. -#figure(image("assets/reidemeister-moves.jpg", height: 7cm)) +#v(1em) + +#figure(image("assets/reidemeister-moves.jpg", height: 9cm)) == Isotopia Regolare @@ -110,7 +114,7 @@ $D$ diagramma di un link orientato, il *writhe* $display(w(D) colon.eq sum_(c "i == Assiomi -Dimostreremo che per ogni diagramma di link _non orientato_ esiste un polinomio $L_D$ per ogni diagramma di link _non orientato_ $D$, il polinomio $L_D in bb(Z)[a, a^(-1), z, z^(-1)]$ verifica i seguenti assiomi: +Dimostreremo che per ogni _diagramma di link non orientato_ $D$, esiste un polinomio $L_D$ con $L_D in bb(Z)[a, a^(-1), z, z^(-1)]$ invariante di _isotopia regolare_ e che verifica i seguenti assiomi: #set enum(numbering: "i.a)") @@ -166,6 +170,8 @@ _Dimostrazione._ $pause #h(3.9em) = F[ #skein.strand-large ]$, [], ) +// SPEAKER NOTE: Per lo stesso diagramma senza quel ricciolo + == Calcoli impliciti #let knot-picture(src, ..rest) = $thin #image("assets/derived/atlas-" + src, ..rest) thin$ @@ -184,7 +190,7 @@ _Dimostrazione._ row-gutter: 1.5em, column-gutter: 3em, align: center, - [*Link $L[#skein.unit-large #skein.unit-large]$*], + [*Calcolo di $L[#skein.unit-large #skein.unit-large]$*], ) ][ #grid( @@ -192,7 +198,7 @@ _Dimostrazione._ row-gutter: 1.5em, column-gutter: 3em, align: center, - [*Link $L[#skein.unit-large #skein.unit-large]$*], + [*Calcolo di $L[#skein.unit-large #skein.unit-large]$*], $ ( space & #knot-picture("infinity-0.png", width: 1.075em) space, @@ -219,7 +225,7 @@ _Dimostrazione._ row-gutter: 1.5em, column-gutter: 3em, align: center, - [*Link $L[#skein.unit-large #skein.unit-large]$*], + [*Calcolo di $L[#skein.unit-large #skein.unit-large]$*], $ & L[ #knot-picture("infinity-0.png", height: 2.25em) ] + @@ -237,7 +243,7 @@ _Dimostrazione._ row-gutter: 1.5em, column-gutter: 3em, align: center, - [*Link $L[#skein.unit-large #skein.unit-large]$*], + [*Calcolo di $L[#skein.unit-large #skein.unit-large]$*], $ & L[ #knot-picture("infinity-0.png", height: 2.25em) ] + @@ -264,7 +270,7 @@ _Dimostrazione._ row-gutter: 1.5em, column-gutter: 3em, align: center, - [*Link $L[#skein.unit-large #skein.unit-large]$*], + [*Calcolo di $L[#skein.unit-large #skein.unit-large]$*], $ & L[ #knot-picture("infinity-0.png", height: 2.25em) ] + @@ -292,7 +298,7 @@ _Dimostrazione._ row-gutter: 1.5em, column-gutter: 3em, align: center, - [*Link $L[#skein.unit-large #skein.unit-large]$*], + [*Calcolo di $L[#skein.unit-large #skein.unit-large]$*], $ & L[ #knot-picture("infinity-0.png", height: 2.25em) ] + @@ -321,7 +327,7 @@ _Dimostrazione._ row-gutter: 1.5em, column-gutter: 3em, align: center, - [*Link $L[#skein.unit-large #skein.unit-large]$*], + [*Calcolo di $L[#skein.unit-large #skein.unit-large]$*], { $ & L[ #knot-picture("infinity-0.png", height: 2.25em) ] @@ -363,7 +369,7 @@ _Dimostrazione._ row-gutter: 1.5em, column-gutter: 3em, align: center, - [*Link $L[#skein.unit-large #skein.unit-large]$*], + [*Calcolo di $L[#skein.unit-large #skein.unit-large]$*], { $ & L[ #knot-picture("infinity-0.png", height: 2.25em) ] @@ -458,6 +464,53 @@ _Dimostrazione._ == Considerazioni preliminari +#slide({ + set align(center) + + [Sia $D$ un diagramma di un link, $i$ etichetta di uno degli incroci:] + + v(1.5em) + + grid( + columns: 4, + column-gutter: 2em, + row-gutter: 1em, + align: center, + { + skein.over-large + place(center + top, dy: -0.75em, { + show math.equation: set text(size: 15pt) + $i$ + }) + }, + { + skein.under-large + place(center + top, dy: -0.75em, { + show math.equation: set text(size: 15pt) + $i$ + }) + }, + { + skein.h-large + place(center + top, dy: -0.75em, { + show math.equation: set text(size: 15pt) + $i$ + }) + }, + { + skein.v-large + place(center + top, dy: -0.75em, { + show math.equation: set text(size: 15pt) + $i$ + }) + }, + + $D$, $S_i D$, $E_i D$, $e_i D$, + ) +}) + +== Considerazioni preliminari + #align(center)[ Il *nodo banale standard* (o in _forma discendente_) associato a $D$ è $hat(D)(cal(U), p)$: @@ -510,53 +563,6 @@ _Dimostrazione._ == Considerazioni preliminari -#slide({ - set align(center) - - [Sia $D$ un diagramma di un nodo, $i$ etichetta di uno degli incroci:] - - v(1.5em) - - grid( - columns: 4, - column-gutter: 2em, - row-gutter: 1em, - align: center, - { - skein.over-large - place(center + top, dy: -0.75em, { - show math.equation: set text(size: 15pt) - $i$ - }) - }, - { - skein.under-large - place(center + top, dy: -0.75em, { - show math.equation: set text(size: 15pt) - $i$ - }) - }, - { - skein.h-large - place(center + top, dy: -0.75em, { - show math.equation: set text(size: 15pt) - $i$ - }) - }, - { - skein.v-large - place(center + top, dy: -0.75em, { - show math.equation: set text(size: 15pt) - $i$ - }) - }, - - $D$, $S_i D$, $E_i D$, $e_i D$, - ) -}) - -== Considerazioni preliminari - #let dotss = $space dots.c space$ #slide( @@ -850,6 +856,8 @@ _Dimostrazione._ // pinit-highlight(1, 2, dy: -1.75em, dx: 3pt, extended-height: 3em, fill: rgb("#0002")) // }) +// SPEAKER NOTE: Scambiamo i segni in modo alternato delle equazioni ottenute. + == Definizione induttiva #slide( @@ -956,7 +964,7 @@ _Dimostrazione._ *Ipotesi induttiva.* Dimostreremo induttivamente le seguenti proprietà: per ogni diagramma di link $D$ con $< N$ incroci e per diagrammi contenenti $#skein.over-twist-large$ con $< N$ incroci: - 1. $L_D$ è ben definito (non dipende dalla scelta di punto base). + 1. $L_D$ è ben definito (non dipende dalla scelta di punto base) 2. $L_D$ verifica gli assiomi: @@ -964,9 +972,9 @@ _Dimostrazione._ - $L[#skein.over-twist-large] = a L [#skein.strand-large]$, $L[#skein.under-twist-large] = a^(-1) L [#skein.strand-large]$ - 3. $L_D$ è invariante per mosse di tipo II e III che non aumentano il numero di incroci. + 3. $L_D$ è invariante per mosse di tipo II e III che non aumentano il numero di incroci - 4. Per certi diagrammi vale $L_D = a^w(D)$. + 4. Per certi diagrammi vale $L_D = a^w(D)$ ] == Dimostrazione buona definizione @@ -992,7 +1000,7 @@ _Dimostrazione._ 3. $L_D$ è invariante per mosse di tipo II e III che non aumentano il numero di incroci - 4. Per certi diagrammi vale $L_D = a^w(D)$. + 4. Per certi diagrammi vale $L_D = a^w(D)$ ] = Laboratorio Computazionale @@ -1019,7 +1027,7 @@ Per il progetto di Laboratorio Computazionale abbiamo scritto una *nuova impleme #pause -- _Trovato un errore nel nodo_ $10_125$: è presente $F[m(10_125)]$ invece di $F[10_125]$ ovvero $F[10_125](1 slash a, z)$. #h(1fr) +- _Trovato un errore nel nodo_ $10_125$: è presente $F[m(10_125)]$ invece di $F[10_125]$ ovvero $F[10_125](1 slash a, z)$ #h(1fr) == Implementazione in Python