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z;|f*vU&epaxWSx%nE`tCuLrY1@z$VI!LQ@LXul%Df2Z;M;VNw0e_j#n&=~UHg=Pox z{N3Mw`zSjn=kMm{f!gL5nA`sd!vPKb{(}Z&`!yr}Z#^IfG)Db*8u$MU6O=#nng#Ow z?npcwY*1tW-k6=|56f_HaP$7I9rTR{wm$@frv3e+KM*?{hy!A50>B0T=J3#YgqQdC z;o{(8`-N!tze;kk^Zd~-bn^JK9yi+`Jh(vI|BNd5-=T(v?gRg@BscW3`kgU!_V|;= z!SjdW+}wW(2t^tA-+G$RQ2;>|`<)5sSD62A8vjBU_=Cpv=P2;-0)Y?#Q@~3yu3syj zs-q)x-G>6;|FZzgI9NDB&)Ff~rT_-6|Nm+sAi$^(U0YCH-Hct_yj;vJP@!`20=a>x Kw6qdRlK&3_|8gb( diff --git a/src/main.typ b/src/main.typ index 126fe92..364e14d 100644 --- a/src/main.typ +++ b/src/main.typ @@ -1156,29 +1156,41 @@ La dimostrazione procede sempre mostrando l'indipendenza dalla scelta di punto b // Questo segue semplicemente sommando le espressioni per $L[K]$ e $L[S_i K]$. #diff-add[ - Dimostriamo ora il seguente lemma nel caso più semplice di aver definito $L_K$ rispetto ad una sola orientazione ovvero, dato $p$ punto base e $0$ indice del primo incrocio nella sequenza di scambi indotta da $p$: + Enunciamo ora il seguente enunciato nel caso di aver definito $L_K$ rispetto ad una sola orientazione ovvero, dato $p$ punto base e $i$ un indice nella sequenza di scambi indotta da $p$: $ L_K (a, z) & colon.eq (-1)^(abs(lambda(p))+1) L[hat(K)(lambda(p))] + z sum_K (lambda(p)) \ - & = z (L[E_0 K] + L[e_0 K]) - L[S_0 K] + & = z (L[E_i K] + L[e_i K]) - L[S_i K] $ - l'ultima uguaglianza segue semplicemente considerando $L[K]$ e $L[S_0 K]$ e sottraendo membro a membro. + l'ultima uguaglianza segue considerando $L[K]$ e $L[S_i K]$ e sottraendo membro a membro. - #lemma[ + #fact[ Se $K$ è un diagramma di link e ammette una funzione di slacciamento allora $L_K = a^w(K)$. ] - #proof[ - Se i punti a valle in $K$ precedono immediatamente i rispettivi punti base allora i diagrammi sono in forma discendente e possiamo utilizzare direttamente la @kauffman-rec-inductive-def di $L_K$ nel caso @kauffman-rec-i. Procediamo per sotto-induzione sul numero totale di incroci e sulla distanza dei punti a valle dai punti base. + Per una dimostrazione di questo fatto si rimanda a @lickorish1997introduction[pag. 175-176]. - Per prima cosa osserviamo che possiamo ignorare gli incroci tra componenti diverse. L'esistenza di una funzione di slacciamento ci induce un ordinamento delle componenti e per il teorema della curva di Jordan tutti i segni degli incroci tra componenti diverse si cancellano. + // #proof[ + // Se i punti a valle in $K$ precedono immediatamente i rispettivi punti base allora i diagrammi sono in forma discendente e possiamo utilizzare direttamente la @kauffman-rec-inductive-def di $L_K$ nel caso @kauffman-rec-i. Procediamo per sotto-induzione sul numero totale di incroci e sulla distanza dei punti a valle dai punti base. - Consideriamo ora una singola componente, sia $i$ il primo incrocio dopo il punto a valle $b$. + // Per prima cosa osserviamo che possiamo ignorare gli incroci tra componenti diverse. L'esistenza di una funzione di slacciamento ci induce un ordinamento delle componenti e per il teorema della curva di Jordan tutti i segni degli incroci tra componenti diverse si cancellano. - - Se partendo da $b$, l'incrocio $i$ è un sotto-incrocio, $h$ può essere modificata - ] + // Consideriamo ora una singola componente, sia $i$ il primo incrocio dopo il punto a valle $b$, ci sono due casi in base a come appare partendo da $b$: + + // - Se è un sotto-incrocio, $h$ può essere modificata in modo da essere decrescente da $b$ fino a subito dopo $i$ in modo che sia ancora una funzione di slacciamento. + + // Dunque $L_K = a^w(K)$ per sotto-induzione in quanto $K$ con la nuova funzione di slacciamento ha il punto $b$ di un incrocio più vicino al punto di base. + + // - Se è un sopra-incrocio deve essere necessariamente un punto in cui una parte della componente in cui andiamo dal punto base verso $b$. Segue dalle proprietà di monotonicità di $h$. + + // In questo caso calcoliamo $L_K$ usando la proprietà $(*)$ applicata all'incrocio $i$. + + // - Scambiare l'incrocio dà un diagramma $K'$ in cui possiamo spostare il punto $b$ più vicino al punto base e dunque per induzione $L_K' = a^w(K')$. + + // - I diagrammi con i due tipi di splice hanno $n-1$ incroci ed ammettono delle funzioni di slacciamento, dunque anche loro sono noti per induzione + // ] #fact[ L'enunciato precedente si generalizza anche con la nostra definizione di $L_K$ che prende in considerazione entrambe le direzioni. La dimostrazione è analoga ma serve tenere conto di come si presentano gli incroci in entrambe le direzioni. @@ -1190,11 +1202,12 @@ La dimostrazione procede sempre mostrando l'indipendenza dalla scelta di punto b 1. Ci sono due casi, in uno otteniamo _due nodi banali in forma standard_, nell'altro _un solo nodo banale_ (non necessariamente in forma discendente). - 2. #diff-del[La proprietà del @std-unknot-to-curls, ovvero che i nodi banali standard sono equivalenti a meno di isotopia regolare a diagrammi di nodi formati solo da riccioli, si estende al diagramma del caso dello splice con una sola componente.] - - #diff-add[ + 2. #diff-add[ Il caso di splice con una sola componente ammette una funzione di slacciamento. ] + + // #diff-del[La proprietà del @std-unknot-to-curls, ovvero che i nodi banali standard sono equivalenti a meno di isotopia regolare a diagrammi di nodi formati solo da riccioli, si estende al diagramma del caso dello splice con una sola componente.] + ] #proof[ diff --git a/src/refs.bib b/src/refs.bib index 95d9df8..6748c9e 100644 --- a/src/refs.bib +++ b/src/refs.bib @@ -44,3 +44,13 @@ year = {1977}, publisher = {Springer New York} } + +@book{lickorish1997introduction, + title = {An Introduction to Knot Theory}, + author = {Lickorish, W.B.R.}, + isbn = {9780387982540}, + lccn = {97016660}, + series = {Graduate Texts in Mathematics}, + year = {1997}, + publisher = {Springer New York} +} \ No newline at end of file diff --git a/src/theme.typ b/src/theme.typ index c341392..91d8302 100644 --- a/src/theme.typ +++ b/src/theme.typ @@ -174,7 +174,7 @@ show raw.where(block: true): it => block(outset: (x: 2pt, y: 3pt), fill: luma(92%), radius: 4pt, inset: 4pt, it) - set std.bibliography(style: "ieee", title: none) + set std.bibliography(style: "ieee", title: none, full: true) set figure(gap: 1em) show figure: set block(above: 1.5em, below: 1.5em)