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z%q%YQ$FK?9nppdEPk5KB^BLU69+`Ft#EfFqwi}N)AZfB_%^!MAN!l z**hU$))us%C=4SH*58Sr_NbmUF<(uFR=JWEIwQ>(51@30PD0P6!$0<5bLG=XzYA$j z?7GbV2#Ta7LR#dkYD)sjIK zrDb7Xc9=jK*`W8uFL-bD%&ne*VO8`R-j?4?7Md= zl$W?dL5ux!4OjppY!xO!D;F+K)6$2Nk_n0Cfj3k6Ul}i+Dq?ws1D)X#jKHomnFQkBaBk~Fct?Qb&ch55Nd zhllg+oq-P=dSS>F{UYAjb8s|T;26_uZuHrI>?T^NrHlAIW Zn_oCKKYa2U3L$nPzh>*!&wqK}{{SigNnii~ diff --git a/src/presentation.typ b/src/presentation.typ index f73d652..7577691 100644 --- a/src/presentation.typ +++ b/src/presentation.typ @@ -44,8 +44,8 @@ // #set enum(indent: 0.6em, body-indent: 0.75em, spacing: 1em, numbering: "i.1.a)") #set par(spacing: 1.5em) -#set list(spacing: 1.5em) -#set enum(numbering: "i.a)", spacing: 1.5em) +#set list(indent: 0.5em, body-indent: 0.75em, spacing: 1.5em) +#set enum(numbering: "i.a)", indent: 0.5em, body-indent: 0.75em, spacing: 1.5em) #show figure.caption: caption => block(inset: (x: 2em), { set align(left) @@ -58,7 +58,7 @@ = Teoria dei Nodi -== Introduzione +== Nodi e Diagrammi // *Def.* $X, Y$ spazi topologici, $f : X arrow Y$ continua è *embedding* se $X approx f(X) subset Y$. @@ -66,9 +66,9 @@ // \ -*Def.* $K subset bb(R)^3$ è un *nodo (tame)* se esiste $f : bb(S)^1 arrow.hook bb(R)^3$ 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 $K = f(bb(S)^1)$. -*Def.* Nodi $bb(S)^1 arrow bb(R)^3 ~>$ *Link* $bb(S)^1 union.sq dots.c union.sq bb(S)^1 arrow bb(R)^3$. +*Link.* $bb(S)^1 arrow bb(R)^3 ~> bb(S)^1 union.sq dots.c union.sq bb(S)^1 arrow bb(R)^3$ #pause @@ -96,9 +96,14 @@ e posta $H_t (x) colon.eq H(x, t)$ si ha: \ -*Esistenza Proiezioni Regolari.* Sia $L subset bb(R)^3$ _link poligonale_, allora esiste un aperto denso $U$ tale che $forall v in U subset SS^2$, la proiezione sul piano $v^perp approx RR^2$ è regolare. +*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*. + + +// di un link $L subset bb(R)^3$ è l'immagine di $L$ attraverso una proiezione decorata con l'informazione sopra/sotto ad ogni incrocio. + +// *Esistenza Proiezioni Regolari.* Sia $L subset bb(R)^3$ _link poligonale_, allora esiste un aperto denso $U$ tale che $forall v in U subset SS^2$, la proiezione sul piano $v^perp approx RR^2$ è regolare. -- $x in pi_v (L) subset v^perp$ tale che $abs(pi^(-1)_v (x)) = 2$ sono detti *incroci*. +// - $x in pi_v (L) subset v^perp$ tale che $abs(pi^(-1)_v (x)) = 2$ sono detti *incroci*. // == Proiezioni e Diagrammi @@ -121,52 +126,61 @@ e posta $H_t (x) colon.eq H(x, t)$ si ha: // ], // ) -== Proiezioni e Diagrammi +== Teorema di Reidemeister -*Def.* 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. -#pad( - top: 1.5em, - grid( - columns: (1fr, auto), - gutter: 1em, - align: horizon, - [ - *Def.* Le mosse I, II, III in figura sono dette *mosse di Reidemeister*. - \ +// #grid( +// columns: (1fr, auto), +// gutter: 1em, +// align: top, +// [ + +// ], +// , +// ) + +*Mosse di Reidemeister.* Le mosse I, II, III in figura sono dette. + +*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)) - *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: 6cm)), - ), -) == Isotopia Regolare +#slide[ -*Def.* $D_1 tilde D_2$ diagrammi di link si dicono equivalenti a meno di *isotopia regolare* se e solo se sono equivalenti a meno di isotopie planari e mosse II e III di Reidemeister. + #set align(center) -\ + #block([ + #set align(left) -#figure(image("assets/ambient-regular-isotopy-difference.jpg", height: 7cm)) + L'*isotopia regolare* è la relazione di equivalenza su diagrammi di link \ generata solo dalle mosse II e III. + ]) + // $D_1 tilde D_2$ diagrammi di link si dicono equivalenti a meno di *isotopia regolare* se e solo se sono equivalenti a meno di isotopie planari e mosse II e III di Reidemeister. -== Comportamento Isotopia Regolare -*Osservazione.* Tutte le mosse di tipo I, quelle che introducono quelli che chiameremo *riccioli*, possono passare sopra o sotto altri fili senza problemi applicando le seguenti mosse + #figure(image("assets/ambient-regular-isotopy-difference.jpg", height: 7cm)) -\ +] -#figure(image("assets/move-1-factorization.png", width: 75%)) +// == Comportamento Isotopia Regolare -== Comportamento Isotopia Regolare +// *Osservazione.* Tutte le mosse di tipo I, quelle che introducono quelli che chiameremo *riccioli*, possono passare sopra o sotto altri fili senza problemi applicando le seguenti mosse -*Osservazione.* Quando abbiamo una coppia di riccioli con segni opposti abbiamo la seguente proprietà di cancellazione detto _trucco di Whitney_. +// \ -\ +// #figure(image("assets/move-1-factorization.png", width: 75%)) -#figure(image("assets/whitney-trick.png", width: 100%)) +// == Comportamento Isotopia Regolare + +// *Osservazione.* Quando abbiamo una coppia di riccioli con segni opposti abbiamo la seguente proprietà di cancellazione detto _trucco di Whitney_. + +// \ + +// #figure(image("assets/whitney-trick.png", width: 100%)) // == Diagrammi in forma discendente @@ -224,33 +238,29 @@ e posta $H_t (x) colon.eq H(x, t)$ si ha: == Writhe -*Def.* Il *segno* di un incrocio di un diagramma orientato è: $epsilon(#skein-generic(size-factor: 1.5, direction: (+1, +1))) = +1, epsilon(#skein-generic(size-factor: 1.5, direction: (+1, -1))) = -1$ +Il *segno* di un incrocio di un diagramma orientato è: $epsilon(#skein-generic(size-factor: 1.5, direction: (+1, +1))) = +1, epsilon(#skein-generic(size-factor: 1.5, direction: (+1, -1))) = -1$ -\ +$D$ diagramma di un link orientato, il *writhe* $display(w(D) colon.eq sum_(c "incrocio") epsilon(c))$. -*Def.* $K$ link orientato, definiamo il *writhe* $display(w(K) colon.eq sum_(c "incrocio") epsilon(c))$. +#figure(image("assets/writhe-examples.png", width: 15cm)) #pause -\ - -*Prop.* Se $K$ è il diagramma di un nodo, il _writhe_ *non dipende dall'orientazione*. +*Prop.* Se $D$ è il diagramma di un nodo, il _writhe_ *non dipende dall'orientazione*. #pause -\ - -*Prop.* Il writhe è un *invariante di isotopia regolare*, ovvero $K_1 tilde K_2 => w(K_1) = w(K_2)$. +*Prop.* Il writhe è un *invariante di isotopia regolare*, ovvero $D_1 tilde D_2 => w(D_1) = w(D_2)$. = Polinomio di Kauffman == Assiomi -*Def.* Per ogni diagramma di link _non orientato_ $K$, il polinomio $L_K in bb(Z)[a, a^(-1), z, z^(-1)]$ verifica i seguenti 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: #set enum(numbering: "i.a)") -1. $K tilde K' => L_K = L_K'$. +1. $D tilde D' => L_D = L_D'$. 2. Valgono le seguenti relazioni: @@ -262,46 +272,49 @@ e posta $H_t (x) colon.eq H(x, t)$ si ha: 4. $L[#skein.under-twist-large] = a^(-1) L[#skein.strand-large]$ -#pause - -\ +// \ -*Osservazione.* A questo punto non sappiamo se $L_K$ _sia ben definito_. +// *Osservazione.* A questo punto non sappiamo se $L_K$ _sia ben definito_. -== Normalizzazione a $F_K$ +== Normalizzazione a invariante di isotopia ambiente -*Def.* $F_K(a, z) colon.eq a^(-w(K)) dot L_K (a, z)$ +*Definizione.* $F_D (a, z) colon.eq a^(-w(D)) dot L_D (a, z)$ #pause -\ +#v(1em) *Prop.* $F_K$ invariante di isotopia ambiente. _Dimostrazione._ -#grid( - columns: (1fr, auto), - align: (left, left, right), - row-gutter: 1.75em, - // stroke: 1pt, +- $w(D)$ inv. isotopia regolare $=> a^(-w(D))$ inv. isotopia regolare. - [ - $F[ #skein.over-twist-large ] = pause a^(-w(#skein.over-twist)) dot L[ #skein.over-twist-large ]$ - ], - { - show math.equation: set text(size: 15pt) - $pause L[#skein.over-twist] = a L[#skein.strand], quad w(#skein.over-twist) = w(#skein.strand) + 1$ - }, +#pause + +- #grid( + columns: (1fr, auto), + align: (left, left, right), + row-gutter: 1.75em, + // stroke: 1pt, + + [ + $F[ #skein.over-twist-large ] = pause a^(-w(#skein.over-twist)) dot L[ #skein.over-twist-large ]$ + ], + { + show math.equation: set text(size: 15pt) + $pause L[#skein.over-twist] = a L[#skein.strand], quad w(#skein.over-twist) = w(#skein.strand) + 1$ + }, - $pause = a^(-w(#skein.strand) - 1) dot a L[ #skein.strand-large ]$, [], + $pause #h(3.9em) = a^(-w(#skein.strand) - 1) dot a L[ #skein.strand-large ]$, [], - $pause = a^(-w(#skein.strand)) cancel(stroke: #1pt, a^(-1)) dot cancel(stroke: #1pt, a) L[ #skein.strand-large ]$, [], + $pause #h(3.9em) = a^(-w(#skein.strand)) cancel(stroke: #1pt, a^(-1)) dot cancel(stroke: #1pt, a) L[ #skein.strand-large ]$, + [], - $pause = a^(-w(#skein.strand)) dot L[ #skein.strand-large ]$, [], + $pause #h(3.9em) = a^(-w(#skein.strand)) dot L[ #skein.strand-large ]$, [], - $pause = F[ #skein.strand-large ]$, [], -) + $pause #h(3.9em) = F[ #skein.strand-large ]$, [], + ) == Calcoli impliciti @@ -619,7 +632,7 @@ _Dimostrazione._ #slide({ set align(center) - [$K$ un diagramma di un nodo, $i$ etichetta di uno degli incroci] + [$D$ un diagramma di un nodo, $i$ etichetta di uno degli incroci] v(1.5em) @@ -661,21 +674,21 @@ _Dimostrazione._ ) - pause + // pause - [\ ] + // [\ ] - [\ ] + // [\ ] - [$lambda = (lambda_n, dots, lambda_0)$ sequenza di etichette di incroci di $K$] + // [$lambda = (lambda_n, dots, lambda_0)$ sequenza di etichette di incroci di $D$] - v(1.5em) + // v(1.5em) - $ - A_i^lambda colon.eq E_lambda_i S_lambda_(i-1) dots.c space S_lambda_0 - #h(2em) - B_i^lambda colon.eq e_lambda_i S_lambda_(i-1) dots.c space S_lambda_0 - $ + // $ + // A_i^lambda colon.eq E_lambda_i S_lambda_(i-1) dots.c space S_lambda_0 + // #h(2em) + // B_i^lambda colon.eq e_lambda_i S_lambda_(i-1) dots.c space S_lambda_0 + // $ }) @@ -695,35 +708,38 @@ _Dimostrazione._ [$p$ punto base direzionato, $lambda$ una sequenza di scambi che porta $K$ a $hat(K)(p)$:] } - show math.equation: set text(size: 18pt) + show math.equation: set text(size: 16pt) - v(1em) + // v(1em) only( - "1-3", + "1-4", $ & L[K] + L[S_0 K] = z( L[E_0 K] + L[e_0 K] ) \ & pause L[S_0 K] + L[S_1 S_0 K] = z( L[E_1 S_0 K] + L[e_1 S_0 K] ) \ + & pause L[S_1 S_0 K] + L[S_2 S_1 S_0 K] = z( L[E_1 S_1 S_0 K] + L[e_1 S_1 S_0 K] ) \ & space dots.v \ & pause L[S_(n-1) dotss S_0 K] + L [hat(K)(p)] = z (L[E_n S_(n-1) dotss S_0 K] + L[e_n S_(n-1) dotss S_0 K]) $, ) only( - 4, + 5, $ & 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_1 S_0 K] + L[e_1 S_0 K] ) \ + & L[S_1 S_0 K] + L[S_2 S_1 S_0 K] = z( L[E_1 S_1 S_0 K] + L[e_1 S_1 S_0 K] ) \ & space dots.v \ (-1)^n ( & L[S_(n-1) dotss S_0 K] + L [hat(K)(p)]) = (-1)^n z (L[E_n S_(n-1) dotss S_0 K] + L[e_n S_(n-1) dotss S_0 K]) $, ) only( - 5, + 6, $ & 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_1 S_0 K] + L[e_1 S_0 K] ) \ + & cancel(L[S_1 S_0 K]) + cancel(L[S_2 S_1 S_0 K]) = z( L[E_1 S_1 S_0 K] + L[e_1 S_1 S_0 K]) \ & space dots.v \ (-1)^n (& cancel(L[S_(n-1) dotss S_0 K]) + L [hat(K)(p)]) = (-1)^n z (L[E_n S_(n-1) dotss S_0 K] + L[e_n S_(n-1) dotss S_0 K]) $, @@ -751,13 +767,14 @@ _Dimostrazione._ [$p$ punto base direzionato, $lambda$ una sequenza di scambi che porta $K$ a $hat(K)(p)$:] } - show math.equation: set text(size: 18pt) + show math.equation: set text(size: 16pt) - v(1em) + // v(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_1 S_0 K] + L[e_1 S_0 K] ) \ + & cancel(L[S_1 S_0 K]) + cancel(L[S_2 S_1 S_0 K]) = z( L[E_1 S_1 S_0 K] + L[e_1 S_1 S_0 K]) \ & space dots.v \ (-1)^n (& cancel(L[S_(n-1) dotss S_0 K]) + L [hat(K)(p)]) = (-1)^n z (L[E_n S_(n-1) dotss S_0 K] + L[e_n S_(n-1) dotss S_0 K]) $ @@ -779,13 +796,14 @@ _Dimostrazione._ [$p$ punto base direzionato, $lambda$ una sequenza di scambi che porta $K$ a $hat(K)(p)$:] } - show math.equation: set text(size: 18pt) + show math.equation: set text(size: 16pt) - v(1em) + // v(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_1 S_0 K] + L[e_1 S_0 K] ) \ + & cancel(L[S_1 S_0 K]) + cancel(L[S_2 S_1 S_0 K]) = z( L[E_1 S_1 S_0 K] + L[e_1 S_1 S_0 K]) \ & space dots.v \ (-1)^n (& cancel(L[S_(n-1) dotss S_0 K]) + L [hat(K)(p)]) = (-1)^n z (L[E_n S_(n-1) dotss S_0 K] + L[e_n S_(n-1) dotss S_0 K]) $ @@ -795,8 +813,10 @@ _Dimostrazione._ h(4.4em) $display( => L[K] + (-1)^n L[hat(K)(p)] = - z sum_(i=0)^n (-1)^i ( - L[A_i^lambda K] + L[B_i^lambda K] + z sum_(i=0)^n (-1)^i lr( + ( + L lr([underbrace(E_i S_(i-1) dotss S_0 K, #place(center, $A_i^lambda$))], size: #1em) + L lr([underbrace(e_i S_(i-1) dotss S_0 K, #place(center, $B_i^lambda$))], size: #1em) + ), size: #1em ) )$ }) @@ -807,13 +827,59 @@ _Dimostrazione._ [$p$ punto base direzionato, $lambda$ una sequenza di scambi che porta $K$ a $hat(K)(p)$:] } - show math.equation: set text(size: 18pt) + show math.equation: set text(size: 16pt) + + // v(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_1 S_0 K] + L[e_1 S_0 K] ) \ + & cancel(L[S_1 S_0 K]) + cancel(L[S_2 S_1 S_0 K]) = z( L[E_1 S_1 S_0 K] + L[e_1 S_1 S_0 K]) \ + & space dots.v \ + (-1)^n (& cancel(L[S_(n-1) dotss S_0 K]) + L [hat(K)(p)]) = (-1)^n z (L[E_n S_(n-1) dotss S_0 K] + L[e_n S_(n-1) dotss S_0 K]) + $ + + v(1.5em) + + h(4.4em) + $display( + => L[K] + (-1)^n L[hat(K)(p)] = + z sum_(i=0)^n (-1)^i (L[A_i^lambda K] + L[B_i^lambda K]) + )$ + + // show math.equation: set text(size: 13pt) + // h(1fr) + // $display( + // (thin #grid( + // rows: 2, + // gutter: 1em, + // $A_i^lambda colon.eq E_lambda_i S_lambda_(i-1) dots.c space S_lambda_0$, + // $B_i^lambda colon.eq e_lambda_i S_lambda_(i-1) dots.c space S_lambda_0$, + // )) + // )$ +}) + +#let hl(bg: gray.mix(white), body) = move(dx: -5pt, dy: 1pt, rect( + fill: bg, + outset: (top: 0.25em, bottom: 0.5em), + radius: 0.25em, + body, +)) + +#slide({ + { + set align(center) + [$p$ punto base direzionato, $lambda$ una sequenza di scambi che porta $K$ a $hat(K)(p)$:] + } + + show math.equation: set text(size: 16pt) - v(1em) + // v(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_1 S_0 K] + L[e_1 S_0 K] ) \ + & cancel(L[S_1 S_0 K]) + cancel(L[S_2 S_1 S_0 K]) = z( L[E_1 S_1 S_0 K] + L[e_1 S_1 S_0 K]) \ & space dots.v \ (-1)^n (& cancel(L[S_(n-1) dotss S_0 K]) + L [hat(K)(p)]) = (-1)^n z (L[E_n S_(n-1) dotss S_0 K] + L[e_n S_(n-1) dotss S_0 K]) $ @@ -823,9 +889,7 @@ _Dimostrazione._ h(4.4em) $display( => L[K] = (-1)^(n+1) L[hat(K)(p)] + - z sum_(i=0)^n (-1)^i ( - L[A_i^lambda K] + L[B_i^lambda K] - ) + z sum_(i=0)^n (-1)^i (L[A_i^lambda K] + L[B_i^lambda K]) )$ }) @@ -835,35 +899,70 @@ _Dimostrazione._ [$p$ punto base direzionato, $lambda$ una sequenza di scambi che porta $K$ a $hat(K)(p)$:] } - show math.equation: set text(size: 18pt) + show math.equation: set text(size: 16pt) - v(1em) + // v(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_1 S_0 K] + L[e_1 S_0 K] ) \ + & cancel(L[S_1 S_0 K]) + cancel(L[S_2 S_1 S_0 K]) = z( L[E_1 S_1 S_0 K] + L[e_1 S_1 S_0 K]) \ & space dots.v \ (-1)^n (& cancel(L[S_(n-1) dotss S_0 K]) + L [hat(K)(p)]) = (-1)^n z (L[E_n S_(n-1) dotss S_0 K] + L[e_n S_(n-1) dotss S_0 K]) $ v(1.5em) + v(2pt) + h(4.4em) $display( - => L[K] = #rect( - fill: gray.mix(white), - outset: (top: 0.25em, bottom: 0.5em), - radius: 0.25em, + => L[K] = + #hl( + bg: white, $display( (-1)^(abs(lambda(q))+1) L[hat(K)(p)] + - z sum_K^text(fill: #gray.mix(white), n) (lambda) + z sum_K^text(fill: #white, n) (lambda) )$, ) )$ - show math.equation: set text(size: 14pt) - h(1fr) - $display((sum_K (lambda) colon.eq sum_(i=0)^n (-1)^i (L[A_i^lambda K] + L[B_i^lambda K])))$ + // show math.equation: set text(size: 14pt) + // h(1fr) + // $display((sum_K (lambda) colon.eq sum_(i=0)^n (-1)^i (L[A_i^lambda K] + L[B_i^lambda K])))$ +}) + +#slide({ + { + set align(center) + [$p$ punto base direzionato, $lambda$ una sequenza di scambi che porta $K$ a $hat(K)(p)$:] + } + + show math.equation: set text(size: 16pt) + + // v(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_1 S_0 K] + L[e_1 S_0 K] ) \ + & cancel(L[S_1 S_0 K]) + cancel(L[S_2 S_1 S_0 K]) = z( L[E_1 S_1 S_0 K] + L[e_1 S_1 S_0 K]) \ + & space dots.v \ + (-1)^n (& cancel(L[S_(n-1) dotss S_0 K]) + L [hat(K)(p)]) = (-1)^n z (L[E_n S_(n-1) dotss S_0 K] + L[e_n S_(n-1) dotss S_0 K]) + $ + + v(1.5em) + + h(4.4em) + $display( + => L[K] = #hl($display( + (-1)^(abs(lambda(q))+1) L[hat(K)(p)] + + z sum_K^text(fill: #gray.mix(white), n) (lambda) + )$) + )$ + + // show math.equation: set text(size: 14pt, fill: white) + // h(1fr) + // $display((sum_K (lambda) colon.eq sum_(i=0)^n (-1)^i (L[A_i^lambda K] + L[B_i^lambda K])))$ }) == Definizione induttiva @@ -875,7 +974,7 @@ _Dimostrazione._ #alternatives[ - *Def (induttiva di $L_K$).* Il polinomio $L_(K)(a,z)$ è definito induttivamente come segue: + *Definizione di $L_K$).* Il polinomio $L_(K)(a,z)$ è definito induttivamente come segue: // la definizione è sotto perché gli enumerate non vanno d'accordo con le figure @@ -910,7 +1009,7 @@ _Dimostrazione._ ][ - *Def (induttiva di $L_K$).* Il polinomio $L_(K)(a,z)$ è definito induttivamente come segue: + *Definizione di $L_K$).* Il polinomio $L_(K)(a,z)$ è definito induttivamente come segue: // la definizione è sotto perché gli enumerate non vanno d'accordo con le figure @@ -989,7 +1088,7 @@ _Dimostrazione._ #set list(spacing: 1.5em) #set enum(numbering: "a)", spacing: 1.5em) - *Def (Ipotesi induttiva).* Dimostreremo induttivamente le seguenti proprietà: per ogni diagramma di link $K$ con $< N$ incroci e per diagrammi contenenti $#skein.over-twist-large$ con $< N$ incroci: + *Ipotesi induttiva.* Dimostreremo induttivamente le seguenti proprietà: per ogni diagramma di link $K$ con $< N$ incroci e per diagrammi contenenti $#skein.over-twist-large$ con $< N$ incroci: 1. $L_K$ è ben definito (non dipende dalla scelta di punto base). @@ -1001,7 +1100,7 @@ _Dimostrazione._ 3. $L_K$ è invariante per mosse di tipo II e III che non aumentano il numero di incroci. - 4. Se $K$ ammette una _funzione di slacciamento_ allora $L_K = a^w(K)$ + 4. Per certi diagrammi vale $L_K = a^w(K)$ ] == Dimostrazione buona definizione @@ -1015,7 +1114,7 @@ _Dimostrazione._ 1. $L_K$ è ben definito (non dipende dalla scelta di punto base) - - Invarianza $sum_K (lambda)$ per $1$-rotazioni + - Invarianza $sum_K (lambda)$ per rotazioni - Caso più componenti @@ -1029,7 +1128,7 @@ _Dimostrazione._ 3. $L_K$ è invariante per mosse di tipo II e III che non aumentano il numero di incroci - 4. Se $K$ ammette una _funzione di slacciamento_ allora $L_K = a^w(K)$ + 4. Per certi diagrammi vale $L_K = a^w(K)$ ] == Dimostrazione buona definizione