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zP+$l_2&yu54sjNb2?CwyVOKu1PIWH9iNE$k))@kS>46T}t>b|M}~Ttt2_ z2)H?6W?LJO(eWorjzuUm2B98+r%M`@K@^4x$_{Z63W_iZF@wk8@1E*cOT_}IdZrTn z-m`QnXn}~rP?33V0^695Pu}Jbng(kAY;euC!$1%Gd7J|ep`!Tn{j~k(%nz#AY+_)8 zp=83g13eW?%!B|28m)PYfb>IzAr$6sp;^D0cn~v0sX>zl-e3@<{y+XiKy{i&OJ`Dv z=10KBna7U>)|e+XgG%UVKxi>Q0w)Rs^34227=(5Wr$y)a9qjFC7;gXHrUrS2u(#<5 z*vG`+LYVaV{sc*M9xX~HKHkvr9)e8~N3cGS5CyrAs2xfpRzQG9C-jLpElR9xbg(~R z6ho7sRja_G8+avQ6A}VmSKxjp&Af~X_6`BpG%P9!Qb7Lylcc7m;`EEVh)DPF$mnn{ TZ}5hQ4pSMZpq!k(rNRFILFnkt diff --git a/src/main.typ b/src/main.typ index 38e213f..4313daf 100644 --- a/src/main.typ +++ b/src/main.typ @@ -648,9 +648,9 @@ Sia ora $lambda = (lambda_n, dots, lambda_0)$ una sequenza di etichette di suoi - Poniamo $hat(K)(lambda) = S_lambda_n dots.c space S_lambda_0 K$ e definiamo le notazioni compatte per le seguenti somme alternate $ - sum_K (lambda) & colon.eq sum_(i=0)^n (-1)^i (L[A_i^lambda K] + L[B_i^lambda K]) \ + sum_K (lambda) & colon.eq sum_(i=0)^n (-1)^i (L[A_i^lambda K] + L[B_i^lambda K]) \ - Omega_K(lambda) & colon.eq (-1)^(abs(lambda) + 1) L[hat(K)(lambda)] + z sum_K (lambda) + Omega_K (lambda) & colon.eq (-1)^(abs(lambda) + 1) L[hat(K)(lambda)] + z sum_K (lambda) $ dove $abs(lambda) = n$ per $lambda$ come sopra. @@ -785,32 +785,36 @@ sempre con $delta = (a + a^(-1)) slash z - 1$. 3. Sia $K = K_1 union dotss union K_n$, se nessun $K_i$ sovrasta tutti gli altri#footnote[Questo sicuramente accade altrimenti potremmo usare il punto ii).] allora: - - Se $K$ è composto da più componenti allora siano $p_1, ..., p_n$ dei punti di partenza direzionati su $K_1, ..., K_n$ e $overline(p)_i$ lo stesso punto di partenza $p_i$ con la direzione opposta 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) - $ - L_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 - - #[ - #set text(size: 11pt) + #[ + #set enum(numbering: "a)") - $ - 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))) - ] - $ - ] + 1. Se $K$ è composto da più componenti allora siano $p_1, ..., p_n$ dei punti di partenza direzionati su $K_1, ..., K_n$ e $overline(p)_i$ lo stesso punto di partenza $p_i$ con la direzione opposta 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) + $ + L_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))) + ] + $ + ] + + 2. 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) + + $ + 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))) + ] + $ + ] + ] == Dimostrazione buona definizione @@ -932,13 +936,69 @@ Nel corso della dimostrazione tutti gli argomenti per induzione si baseranno sul riordiniamo le operazioni e portiamo $E_0$ e $e_0$ all'inizio delle successioni di operazioni + #let hl(c: color.mix((white, 100%), (black, 10%)), content) = box(radius: 2pt, outset: (y: 3pt, x: 1pt), fill: c, content) + $ - = + ( & L[E_0 K] + L[e_0 K]) + \ - - z( & L[ E_1 E_0 K] + L[ E_1 e_0 K] + L[ e_1 E_0 K] + L[ e_1 e_0 K] ) + \ + = + ( & L[#hl($E_0 K$)] + L[#hl($e_0 K$)]) + \ + - z( & L[ E_1 #hl($E_0 K$)] + L[ E_1 #hl($e_0 K$)] + L[ e_1 #hl($E_0 K$)] + L[ e_1 #hl($e_0 K$)] ) + \ space dots.v \ - (-1)^n z ( & L[ E_n S_(n-1) dotss S_1 E_0 K ] + L[ E_n S_(n-1) dotss S_1 e_0 K ] + \ - & + L[ e_n S_(n-1) dotss S_1 E_0 K ] + L[ e_n S_(n-1) dotss S_1 e_0 K ] ) \ - + (-1)^(n+1) ( & L[ S_n dotss S_1 E_0 K] + L[ S_n dotss S_1 e_0 K] ) + (-1)^n z ( & L[ E_n S_(n-1) dotss S_1 #hl($E_0 K$) ] + L[ E_n S_(n-1) dotss S_1 #hl($e_0 K$) ] + \ + & + L[ e_n S_(n-1) dotss S_1 #hl($E_0 K$) ] + L[ e_n S_(n-1) dotss S_1 #hl($e_0 K$) ] ) \ + + (-1)^(n+1) ( & L[ S_n dotss S_1 #hl($E_0 K$)] + L[ S_n dotss S_1 #hl($e_0 K$)] ) + $ + + $ + = L[ #hl($E_0 K$)] &+ (-1)^(n+1) L[S_n dotss S_1 #hl($E_0 K$)] + \ + - z lr(size: #1.5em, ( ( & L[E_1 #hl($E_0 K$)] + L[e_1 #hl($E_0 K$)])+ \ + - ( & L[E_2 S_1 #hl($E_0 K$)] + L[e_2 S_1 #hl($E_0 K$)]) +\ + & space dots.v \ + +( -1)^n ( & L[E_n S_(n-1) dots S_1 #hl($E_0 K$)] + L[e_n S_(n-1) dots S_1 #hl($E_0 K$)]))) \ + + L[ #hl($e_0 K$)] &+ (-1)^(n+1) L[S_n dotss S_1 #hl($e_0 K$)] + \ + - z lr(size: #1.5em, ( ( & L[E_1 #hl($e_0 K$)] + L[e_1 #hl($e_0 K$)])+ \ + - ( & L[E_2 S_1 #hl($e_0 K$)] + L[e_2 S_1 #hl($e_0 K$)]) +\ + & space dots.v \ + +( -1)^n ( & L[E_n S_(n-1) dots S_1 #hl($e_0 K$)] + L[e_n S_(n-1) dots S_1 #hl($e_0 K$)]))) + $ + + Infine osserviamo che i termini in "z ( #blank )" sono proprio le somme incrementali sugli scambi per $E_0 K$ e $e_0 K$, dunque possiamo riscrivere l'espressione precedente come + + $ + = L[ E_0 K] &+ (-1)^(n+1) L[S_n dotss S_1 E_0 K] - z sum_(E_0 K) ((n, dots, 1)) \ + + L[ e_0 K] &+ (-1)^(n+1) L[S_n dotss S_1 e_0 K] - z sum_(e_0 K) ((n, dots, 1)) + $ + + ora possiamo applicare l'ipotesi induttiva in quanto tutti i diagrammi hanno strettamente meno di $N$ incroci. Applicando l'ipotesi induttiva per $gamma = (n, dots, 1)$ e osservando che con la convenzione di Kauffman $abs(gamma) + 1 = (n - 1) + 1 = n$, otteniamo che + $ + Omega_K(gamma) + &= (-1)^(abs(gamma) + 1) L_(hat(K) (gamma)) + z sum_K (gamma) \ + &= (-1)^(n+1) L_(hat(K) (gamma)) - z sum_K (gamma) + $ + + infine sostituendo + + $ + &= L[E_0 K] - Omega_(E_0 K) (gamma) + L[e_0 K] - Omega_(e_0 K) (gamma) \ + &= 0 + 0 = 0 + $ + + E questo conclude la dimostrazione. +] + +*Corollario.* Nella definizione della forma chiusa di $L_K$, il caso ii.a) non dipende dalla scelta del punto base. + +#proof[ + Rivediamo l'espressione in questione + + $ + L_K (a, z) colon.eq + 1 / (2n) [ + sum_(i=1)^n sum_(q=p_i, overline(p)_i) + ((-1)^(abs(lambda(q))+1) d kL_(K_i) kL_(K - K_i) + z sum_K (lambda(q))) + ] + $ + + se applichiamo una $1$-rotazione alle sequenze di scambi le lunghezze non cambiano ed abbiamo appena mostrato che $sum_K (lambda(q)))$ è invariante per $1$-rotazioni. + A questo punto possiamo estendere l'invarianza per $1$-rotazioni a tutte le possibili permutazioni cicliche delle sequenze di scambi e questo ci dimostra che questo caso non dipende dalla scelta del punto base. ] \ No newline at end of file diff --git a/src/prelude.typ b/src/prelude.typ index ee362de..260cfdf 100644 --- a/src/prelude.typ +++ b/src/prelude.typ @@ -27,37 +27,28 @@ #let definition(body, name: none) = { // show figure: statement-style(name, numbered: false) - figure( - body, - kind: "definition", - supplement: { - [Definizione] - if name != none { - [ -- ] - name - } - }, - ) + figure(body, kind: "definition", supplement: { + [Definizione] + if name != none { + [ -- ] + name + } + }) } #let fact(body, name: none) = { // show figure: statement-style(name, numbered: false) - figure( - body, - kind: "fact", - supplement: { - [Fatto] - if name != none { - [ -- ] - name - } - }, - ) + figure(body, kind: "fact", supplement: { + [Fatto] + if name != none { + [ -- ] + name + } + }) } #let proposition(body, numbered: true) = { // show figure: statement-style(name, numbered: numbered) - return figure( body, // kind: "proposition", @@ -80,17 +71,12 @@ #let theorem(body, name: none, numbered: true) = { // show figure: statement-style(name, numbered: numbered) - figure( - body, - kind: "theorem", - supplement: { - [Teorema] - if name != none { - [ (#name)] - } - }, - numbering: if numbered { "1" }, - ) + figure(body, kind: "theorem", supplement: { + [Teorema] + if name != none { + [ (#name) ] + } + }, numbering: if numbered { "1" }) } #let proof(body) = block({ @@ -125,34 +111,23 @@ } #let todo-color = color.mix((red, 10%), (yellow, 90%)) -#let todo(content) = block( - breakable: false, - grid( - rows: 2, - align: left, - block( - fill: todo-color.desaturate(60%), - inset: (x: 0.5em, y: 0.35em), - radius: (top: 0.25em), - { - set text(fill: black.transparentize(15%), size: 8pt, font: "Open Sans") - [*TODO*] - }, - ), - block( - width: 100%, - fill: todo-color.desaturate(75%), - inset: (x: 0.5em, y: 0.5em), - radius: (bottom: 0.25em, top-right: 0.25em), - { - set text(fill: black.transparentize(15%), size: 9pt, font: "Open Sans") - content - }, - ), +#let todo(content) = block(breakable: false, grid( + rows: 2, + align: left, + block(fill: todo-color.desaturate(60%), inset: (x: 0.5em, y: 0.35em), radius: (top: 0.25em), { + set text(fill: black.transparentize(15%), size: 8pt, font: "Open Sans") + [*TODO*] + }), + block( + width: 100%, + fill: todo-color.desaturate(75%), + inset: (x: 0.5em, y: 0.5em), + radius: (bottom: 0.25em, top-right: 0.25em), + { + set text(fill: black.transparentize(15%), size: 9pt, font: "Open Sans") + content + }, ), -) +)) -#let scr(it) = text( - features: ("ss01",), - box($cal(it)$), -) +#let scr(it) = text(features: ("ss01",), box($cal(it)$))