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ze(<1PAE_S(NbgI#YU^>jfTz9Q*SovQbsxUoRlDp7Kcme&v8RsBez5x7p1QQUNZQvw zo!;L+7#+fwUH!$LIz6CRf4`@muE^QIiu2#_;K^nD195HS{F}l8UIpcV&S4>P2Fzdu zT3Cr!c#PkZctwGHSLK^at+~O?2$l{aNfy;{!+a@C4I4NsgvQZX zTuYwWS7qgAmxY81{%f4ecK~K#kc=D9yu<6gymjsr3=5pIGnnSmDr!j6is|S`vgq%gv?i$%qkA5+p%< z@87cmC2>6qYn(?6J>M>46cGoPMZ^h3Dl{U7`_)uq6MPgQht0?xLO>j5DgE6V>!l({ zZQkbLuQ{p_c3EcBj!cv$bmSphE}$=Q@WQM_6tz&K2nZ4(bP*7HbMqfGnUSA%5O=ky z6%g6Pvk)>E_;TxYQ==JbrI2=Sgfk}(vW(8QS18<7_b08!)s zgYmIiTaUz$({tY1 zq-IpJQVidBvKHW&Ob8&OzGBiQWbm zGJ)r{@M;kBR#r*Cd|yCMv;4vKpPL_uu~)|_;7lcBxV3ONOdW>7dU!~M0#@K+kywWL zn^2%e2`7Z3mp#UX*Dj9|>@Cn43MQpCeEsCVPPx*}p8Dl5s(j)MGhaX@={X$yvJ`P_ z0!kBi@EO4vLeGL|j%0YEOq<3(p#NM#(wGq{;*$_ee4kC302buAqZ2Ab)-$LtB>jjC z+drWP{y>Be3rzqEd@IcuTZ5RFu%M+cpkoTsS&?O|%qg6CDt=s&Gl{5AJs1u^z??SH zXP`#G-FQaNp}r&zhc)iJO>3=cVSZ!_8SjWhC;9+vM&~1s2@k17lf`(l0by8)M 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)) - $ - - quindi passare al mirror ha l'effetto di scambiare solamente $a$ con $a^(-1)$ ma il resto della valutazione rimane identico. - - Infine per $F_m(K)$ basta osservare che $w(m(K)) = -w(K)$ per una motivazione analoga. -] +// #proof[ +// Segue dalla @prop-ext-ambient-isotopy-inv. +// ] -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 -$ - - -E 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[ - 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 $K = K_1 hash K_2$. -] +== Alcune proprietà del polinomio di Kauffman -#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. -] +#todo[ + work in progress +] + +// #lemma[ +// Sia $K$ un link orientato e $m(K)$ il suo mirror. Allora abbiamo la seguente relazione per $L$ +// e $F$ + +// $ +// L_(m(K)) (a, z) = L_K (1 slash a, z) +// #h(2em) +// F_(m(K)) (a, z) = F_K (1 slash a, z) +// $ + +// ] + +// #proof[ +// $m(K)$ è ottenuto da $K$ scambiando tutti i suoi incroci 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) +// $ + +// 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)) +// $ + +// quindi passare al mirror ha l'effetto di scambiare solamente $a$ con $a^(-1)$ ma il resto della valutazione rimane identico. + +// Infine per $F_m(K)$ basta osservare che $w(m(K)) = -w(K)$ per una motivazione analoga. +// ] + +// 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 +// $ + + +// E 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[ +// 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. +// ] == Dimostrazione Forma Induttiva -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 $cal(U)$. +#todo[ + work in progress ] -- 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. +// Prima di passare a descrivere la definizione induttiva diamo alcune definizioni di manipolazioni di un diagramma. - #figure( - image( - "assets/standard-unlink-construction.png", - width: 12cm, - ), - caption: [ - Un link con tre componenti e punti di partenza direzionati $p_1, p_2, p_3$, poniamo $overline(p) = (p_1, p_2, p_3)$. In $hat(L)(cal(U), overline(p))$ sono evidenziati in #box(radius: 3pt, outset: (y: 3pt), inset: (x: 2pt), fill: rgb("#91cdff"))[azzurro] gli incroci \ scambiati rispetto al link $L$. - ], - ) +// #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. -- Sia $K$ un link, $lambda = (lambda_n, dots, lambda_0)$ una sequenza di etichette di incroci allora poniamo +// #figure( +// image( +// "assets/standard-unlink-construction.png", +// width: 12cm, +// ), +// caption: [ +// Un link con tre componenti e punti di partenza direzionati $p_1, p_2, p_3$, poniamo $overline(p) = (p_1, p_2, p_3)$. In $hat(L)(cal(U), overline(p))$ sono evidenziati in #box(radius: 3pt, outset: (y: 3pt), inset: (x: 2pt), fill: rgb("#91cdff"))[azzurro] gli incroci \ scambiati rispetto al link $L$. +// ], +// ) - $ - 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. +// - Sia $K$ un link, $lambda = (lambda_n, dots, lambda_0)$ una sequenza di etichette di incroci allora poniamo -- #todo[Operazione compatta $sum_K (lambda)$] +// $ +// 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 +// $ -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 +// 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. -$ - 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)) -$ +// - #todo[Operazione compatta $sum_K (lambda)$] -se ora sommiamo e sottraiamo membro a membro otteniamo la seguente identità +// 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 -#context block(width: page.width, grid( - columns: 3, - column-gutter: 2em, - row-gutter: 1em, - $L_K + L_(S_0 K)$, [], $z( L_(E_0 K) + L_(e_0 K) )$, - $-(L_(S_0 K) + L_(S_1 S_0 K))$, [], $-z( L_(E_0 S_0 K) + L_(e_0 S_0 K) )$, - $dots.v$, $=$, $dots.v$, - $+(-1)^n (L_(S_(n-1) dots S_0 K) + L_(hat(K)))$, - [], - $+(-1)^n (z (L_(E_n S_(n-1) dots S_0 K) + L_(e_n S_(n-1) dots S_0 K)))$, -)) +// $ +// 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)) +// $ -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, - $L_K + cancel(L_(S_0 K))$, [], $z( L_(E_0 K) + L_(e_0 K) )$, - $-(cancel(L_(S_0 K)) + cancel(L_(S_1 S_0 K)))$, [], $-z( L_(E_0 S_0 K) + L_(e_0 S_0 K) )$, - $dots.v$, $=$, $dots.v$, - $+(-1)^n (cancel(L_(S_(n-1) dots S_0 K)) + L_(hat(K)))$, - [], - $+(-1)^n (z (L_(E_n S_(n-1) dots S_0 K) + L_(e_n S_(n-1) dots S_0 K)))$, -)) +// se ora sommiamo e sottraiamo membro a membro otteniamo la seguente identità -Da cui otteniamo un'espressione per $L_K$, inoltre posto $lambda = (0, dots, n)$ +// #context block(width: page.width, grid( +// columns: 3, +// column-gutter: 2em, +// row-gutter: 1em, +// $L_K + L_(S_0 K)$, [], $z( L_(E_0 K) + L_(e_0 K) )$, +// $-(L_(S_0 K) + L_(S_1 S_0 K))$, [], $-z( L_(E_0 S_0 K) + L_(e_0 S_0 K) )$, +// $dots.v$, $=$, $dots.v$, +// $+(-1)^n (L_(S_(n-1) dots S_0 K) + L_(hat(K)))$, +// [], +// $+(-1)^n (z (L_(E_n S_(n-1) dots S_0 K) + L_(e_n S_(n-1) dots S_0 K)))$, +// )) -#align(center, block(width: 15cm, [ - #set align(center) +// notiamo che possiamo cancellare tutti i termini del membro di sinistra che compaiono due volte con segno opposto - $ - => - 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) - ) - $ +// #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 $lambda = (0, dots, n)$ -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. +// #align(center, block(width: 15cm, [ +// #set align(center) -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)^(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) +// ) +// $ -$ - L_(K_1 union.sq K_2) = delta L_(K_1) L_(K_2) -$ +// ])) -sempre con $delta = (a + a^(-1)) slash z - 1$. +// 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. -// Closed form algorithm +// 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 $K_2$. -#definition(name: [Forma chiusa per $kL_K$])[ - Il polinomio $kL_(K)(a,z)$ è definito induttivamente come segue, abbiamo i seguenti casi: -] +// $ +// L_(K_1 union.sq K_2) = delta L_(K_1) L_(K_2) +// $ -// la definizione è sotto perché gli enumerate non vanno d'accordo con le figure +// sempre con $delta = (a + a^(-1)) slash z - 1$. -1. Se $K = hat(K)$ è un _nodo banale standard_ allora $kL_K (a, z) colon.eq a^w(K)$ +// // Closed form algorithm -2. Se $K_1$ è _sovrastante_ $K_2$, sia $d colon.eq (a + a^(-1)) slash z - 1$ e allora +// #definition(name: [Forma chiusa per $kL_K$])[ +// Il polinomio $kL_(K)(a,z)$ è definito induttivamente come segue, abbiamo i seguenti casi: +// ] - $ - kL(K_1 union.sq K_2) colon.eq d kL(K_1) kL(K_2) - $ +// // 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 $kL_K (a, z) colon.eq a^w(K)$ -3. Se $K = K_1 union.sq dotss union.sq K_n$ +// 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) +// $ - - Se un $K_i$ è _sovrastante_ un'altra componente allora applica (ii). +// 3. Se $K = K_1 union.sq dotss union.sq K_n$ - - 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 un $K_i$ è _sovrastante_ un'altra componente allora applica (ii). - #[ - #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 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 $K$ 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 / (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) +// - 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 - $ - 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))) - ] - $ - ] +// #[ +// #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))) +// ] +// $ +// ]

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