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zDg)4fg8Gw&`-OfK5&?t$BLo5Xny@hfQ2ZYn=AQusz$%6yoG4)52K?`~I0!(5|I--o z^*>$!3fGU437GPq(nmNVR1g>$AcUaNB>-6e5sU%6Pk?k327$u- z1tcI$7!=@tg8ndA1eDMV7%b$!kqh^~NdgKBM_~zy1quuJh7ccEEEe$(ANI%by?@0R ziv>{qw=#fZ(1e2>4h_NlGx!7L<6jyC3y=vr1a{XDfD2%P5&{#T0oD<;g+c)h!9T~J zU-1OSWhnBW4q;GW+s|JJ2KW&F3PE6xW~P670SzP~FeN3V0fWRZ8t&H|4UU2nS_hN> z3;~dU4zYysL_-k}!a5Jo{uvJd8UjKX61}4d!++ZRQU*|v09?Qf0w`3!wEo41{Acol fh60$0H4p_^{Xa9>9#Ie@HS`GlR00Buno9o%3%fTp diff --git a/src/main.typ b/src/main.typ index 673d6e8..1f17b4a 100644 --- a/src/main.typ +++ b/src/main.typ @@ -498,19 +498,17 @@ Prima di passare alla dimostrazione della buona definizione vediamo alcuni esemp 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] -#figure(image("assets/implicit-calc-1.png")) - -#let knot-picture(src) = $thin #image("assets/derived/atlas-" + src, width: 1.125em) thin$ +#let knot-picture(src, ..rest) = $thin #image("assets/derived/atlas-" + src, ..rest) thin$ $ - L( #knot-picture("infinity-0.png") ) + L( #knot-picture("infinity-0.png", height: 2.25em) ) + - L( #knot-picture("infinity-1.png") ) + L( #knot-picture("infinity-1.png", height: 2.25em) ) &= z ( - L( #knot-picture("infinity-2.png") ) + L( #knot-picture("infinity-2.png", height: 2.25em) ) + - L( #knot-picture("infinity-3.png") ) + L( #knot-picture("infinity-3.png", height: 2.25em) ) ) \ => a L( #skein.unit ) @@ -526,20 +524,73 @@ $ => 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 valore del link di Hopf come segue +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 +$ -#figure(image("assets/implicit-calc-2.png")) Ed del nodo trifoglio -#figure(image("assets/implicit-calc-3.png")) +// #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_ $tilde(K) = K_1 union.sq K_2$, in questo caso chiamiamo la somma connessa $K = K_1 hash K_2$. + 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[ @@ -602,6 +653,8 @@ se ora sommiamo e sottraiamo membro a membro otteniamo la seguente identità ), ) +notiamo che possiamo cancellare tutti i termini del membro di sinistra che compaiono due volte con segno opposto + #context block( width: page.width, grid( @@ -617,7 +670,7 @@ se ora sommiamo e sottraiamo membro a membro otteniamo la seguente identità ), ) -Da cui otteniamo che +Da cui otteniamo un'espressione per $L_K$, inoltre posto $lambda = (0, dots, n)$ #align( center, @@ -627,17 +680,25 @@ Da cui otteniamo che #set align(center) $ - => L_K = (-1)^(n+1) L_(hat(K)) + z sum_(i=0)^n (-1)^i ( + => + 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) ) $ + ], ), ) 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. -Affinché sia sempre possibile raggiungere un nodo banale in forma standard serve aggiungere anche la seguente relazione quando abbiamo $K = K_1 union.sq K_2$ con $K_1$ che sovrasta $K_2$ +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 union.sq K_2) = delta L_(K_1) L_(K_2) @@ -648,7 +709,7 @@ sempre con $delta = (a + a^(-1)) slash z - 1$. // Closed form algorithm #definition(name: [Forma chiusa per $kL_K$])[ - Il polinomio $kL_(K)(a,z)$ è definito induttivamente con i seguenti casi + Il polinomio $kL_(K)(a,z)$ è definito induttivamente come segue, abbiamo i seguenti casi: ] // la definizione è sotto perché gli enumerate non vanno d'accordo con le figure diff --git a/src/theme.typ b/src/theme.typ index ac710c3..cabed07 100644 --- a/src/theme.typ +++ b/src/theme.typ @@ -450,30 +450,58 @@ } // Display the title and authors. - v(35pt, weak: true) - align( - center, - { - text(size: 18pt, weight: 600, title) - v(25pt, weak: true) - smallcaps({ - text(author-string) - }) - }, - ) + // v(35pt, weak: true) + // align( + // center, + // { + // text(size: 18pt, weight: 600, title) + // v(25pt, weak: true) + // smallcaps({ + // text(author-string) + // }) + // }, + // ) // Display the abstract if abstract != none { - v(20pt, weak: true) // set text(script-size) - show: pad.with(x: 35pt) + // show: pad.with(x: 2em) - set text(size: 11pt) + // set align(center + horizon) - [*Abstract.* ] - abstract + pad( + x: 2em, + grid( + columns: 1, + v(6em), + { + set align(center) + [*Abstract:*] + }, + + v(1em), + { + set align(start) + abstract + }, + + v(2em), + ), + ) + + // { + // set align(center) + + // [*Abstract*] + + // abstract + // } } + // pagebreak() + + // layout(size => v(size.height * 25% - 6pt)) + outline(title: "Indice", depth: 2, indent: 1em) context { counter("fact").update(2) }