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zz<}{shTsEWWE?oitza4$I*8z_U|9MgZMkp&55Fv8F60BiGKs4Plb76NAyWcajGGBy zhKK`(#4rh%Wu<%4@kk7w17K7lg??OI$sq~DxCIl4AkSh@KY*=Dfv&YK8U^`$`AsG!a2L?P~mW(z7a;H_s3ie50Bmc(I=f?n7 z{6F~}Fm*<*frHTxTq_u)0O1%J1b)1VX$l3iEDu~v`KUxjf^^Se^lc)V0Ft?t6Vr{F z(LIT15{X{^Sk55G4d>3eD`7+e{kfny{V%GYhy_|;SYrUgts;m8**YULSg_lz8cU`y zBrh0{SAsQJ#vsr?$W!qkxGn?E!Eh7;BS9Q~uA03<5*%11MrLpnEW=I-<^ZvWQ8Rcv zn#yQ2Af3f9`a2OsTZU8$0}|C$E(Sai$7l;gB6ihz0(c^N!4{VCKM=k_;c4am4q$k~ zs?;1DSr`KikpyDlsa29y^> p*f(q<{Qr`Hy87n1=ZAcJt-XEyz3m-1z+@N-3Bw^PtGj2%{{Rpj#VG&) diff --git a/src/main.typ b/src/main.typ index 64c1ec8..043bb77 100644 --- a/src/main.typ +++ b/src/main.typ @@ -860,7 +860,7 @@ Nel corso della dimostrazione tutti gli argomenti per induzione si baseranno sul $ sum_K (lambda) = sum_K (mu) $ -] +] #proof[ Per prima cosa rinominiamo le etichette $lambda_n, dots, lambda_0$ come $n, dots, 0$ per semplificare la notazione dunque @@ -914,16 +914,6 @@ Nel corso della dimostrazione tutti gli argomenti per induzione si baseranno sul + (-1)^(n+1) ( & L[E_0 S_n dotss S_1 K] + L[e_0 S_n dotss S_1 K] ) $ - #let blank = { - $#h(0.125em) #{ - rect(width: 0.8em * 1.41, height: 0.8em, stroke: 0.5pt, place(dx: -0.2em, dy: -0.55em, { - set text(size: 10pt) - - $dots$ - })) - } #h(0.125em)$ - } - Permutando i termini di tutte le righe tranne la prima e l'ultima possiamo riscriverli in modo da averli nella forma $L[#blank K] + L[#blank S_0 K]$: $ @@ -1055,7 +1045,7 @@ $ 1. $S_i hat(K) = hat(K)'$ 2. $L[hat(K)] + L[hat(K)'] = z (L[E_i hat(K)] + L[e_i hat(K)])$ -] +] #proof[ 1. Per definizione di nodo banale standard $hat(K)(cal(U), q) = S_i hat(K)(cal(U), p)$ poiché l'incrocio $i$ in $hat(K)(cal(U), p)$ compare come sopra-incrocio mentre in $hat(K)(cal(U), q)$ è l'ultimo incrocio visitato quindi è certamente la seconda occorrenza di $i$ quindi è un sotto-incrocio. Tutti gli altri incroci compaiono nello stesso ordine quindi restano invariati. @@ -1076,7 +1066,7 @@ $ L[hat(K)] &= a^(w+1) \ L[S_i hat(K)] &= a^(w-1) \ L[e_i hat(K)] &= a^w \ - L[E_i hat(K)] &= d a^(w_1) a^(w_2) = d a^(w_1 + w_2) = d a^w = (a^(w+1) + a^(w-1))/z - a^w + L[E_i hat(K)] &= d a^(w_1) a^(w_2) = d a^(w_1 + w_2) = d a^w = (a^(w+1) + a^(w-1)) slash z - a^w $ che dunque verifica l'identità: @@ -1093,6 +1083,145 @@ $ Sia $K$ un diagramma di un nodo. Allora ii.b) non dipende dalla scelta di punto base e più precisamente vale $ - Omega_K(p) = (-1)^(abs(lambda(p)) + 1) + Omega_K(p) = (-1)^(abs(lambda(p)) + 1) L_hat(K)(p) + z sum_K (lambda(p)) $ +] + +#proof[ + Possiamo assumere che la sequenza di scambi determinata da $p$ sia etichetta $lambda = (n, dots, 0)$. Mostriamo l'indipendenza da punto base mostrando che possiamo scorrere $p$ nell'arco successivo al primo incrocio dopo $p$. + + Sia $i$ l'etichetta del primo incrocio dopo $p$. In $hat(K)(p)$ questo incrocio sarà sicuramente un sopra-incrocio in quanto $i$ è visitato per la prima volta essendo il primo incrocio dopo $p$. Invece in $K(p)$ può essere sia un sopra-incrocio che un sotto-incrocio, abbiamo quindi due casi: + + - Se $i$ è un sotto-incrocio in $K(p)$ allora $i$ sarà il primo incrocio nella sequenza di scambi ed avremo $i = n$ + + #todo[disegnino] + + Consideriamo ora la situazione di $K(q)$ + + #todo[disegnino] + + come già detto prima se l'ultimo incrocio è un sotto-incrocio allora è già in forma di nodo banale standard. Segue che $i$ non appartiene alla sequenza di scambi in questo caso ed avremo $(n-1, dots, 0)$ e $hat(K)(q) = S_(n-1) dotss S_0 K$. Vorremo vedere che $Omega_K (p) = Omega_K (q)$, ovvero: + + $ + Omega_K (p) &= (-1)^(n+1) L[hat(K)(p)] + z sum_K (lambda(p)) \ + Omega_K (q) &= (-1)^((n-1)+1) L[hat(K)(q)] + z sum_K (lambda(q)) + $ + + come in precedenza studiamo la differenza: + + $ + Omega_K (p) - Omega_K (q) + =& (-1)^(n+1) L[hat(K)(p)] + z sum_K (lambda(p)) \ + &+ underbrace((-1)^(n+1), = -(-1)^n) L[hat(K)(q)] - z sum_K (lambda(q)) \ + =& (-1)^(n+1) (L[hat(K)(p)] + L[hat(K)(q)]) \ + &+ z (-1)^n (L[A_n^lambda K] + L[B_n^lambda K]) + $ + + dove abbiamo usato il fatto che tutti i termini in $sum_K (lambda(p)) - sum_K (lambda(q))$ si cancellano tra loro tranne l'ultimo di $sum_K (lambda(p))$. + + Ora notiamo che + + $ + A_n^lambda K &= E_n S_(n-1) dotss S_0 K = E_n hat(K)(p) \ + B_n^lambda K &= e_n S_(n-1) dotss S_0 K = e_n hat(K)(p) \ + hat(K)(p) &= S_n space.med hat(K)(p) + $ + + inoltre per il @lemma-slide-identities abbiamo anche che + + $ + L[hat(K)(p)] + L[hat(K)(q)] = z (L[E_n hat(K)(p)] + L[e_n hat(K)(p)]) + $ + + Per concludere questo caso basta sostituire in queste ultime identità + + $ + Omega_K (p) - Omega_K (q) =& (-1)^(n+1) (L[hat(K)(p)] + L[hat(K)(q)]) \ + &+ z (-1)^n (L[A_n^lambda K] + L[B_n^lambda K]) \ + = & cancel(z (-1)^(n+1) (L[E_n hat(K)(p)] + L[e_n hat(K)(p)])) \ + &+ cancel(z (-1)^n (L[E_n hat(K)(p)] + L[e_n hat(K)(p)])) \ + = & 0 + $ + + e quindi $Omega_K (p) = Omega_K (q)$. + + - Se $i$ è un sopra-incrocio in $K(p)$ siamo nella seguente situazione + + #todo[disegnino] + + In questo caso, $i$ non fa parte della sequenza di scambi per $K(q)$. Però dopo aver spostato il punto base avremo la seguente situazione + + #todo[disegnino] + + Questa volta $i$ fa parte della sequenza di scambi per $K(q)$ e quando compare come ultimo incrocio è un sotto-incrocio. Avremo quindi le seguenti sequenze di scambi + + $ + lambda(q) &= (n, n-1, dots, i, dots, 1, 0) \ + lambda(p) &= (n, n-1, dots, i+1, i-1, dots, 1, 0) + $ + + a meno di riordinare gli scambi è facile vedere che $S_i hat(K)(q) = hat(K)(p)$, e che il @lemma-slide-identities si applica alla coppia $hat(K)(p), hat(K)(q)$ per l'indice $i$. Per il @lemma-sum-switches-rotation abbiamo anche che $sum_K (p), sum_K (q)$ sono invarianti a meno di permutazioni cicliche. Possiamo quindi sostituire $lambda(p)$ e $lambda(q)$ con le seguenti sequenze di scambi applicando le giuste rotazioni. + + $ + lambda'(p) &= (i-1, dots, 1, 0, n, n-1, dots, i+1) \ + lambda'(q) &= (i, i-1, dots, 1, 0, n, n-1, dots, i+1) + $ + + A questo punto possiamo applicare un argomento simile a quello del punto precedente. + + E questo completa la dimostrazione dell'indipendenza di $Omega_K (p)$ dalla scelta di punto base. +] + +#lemma[ + Sia $i$ un incrocio di un diagramma di link $K$. Allora $L_K$ verifica le identità + + 1. $L[K] + L[S_i K] = z (L[E_i K] + L[e_i K])$ + + 2. $L[#skein.over-twist-medium] = a L [#skein.strand-medium]$, $L[#skein.under-twist-medium] = a^(-1) L [#skein.strand-medium]$ +] + +#proof[ + Mostriamo per induzione sul numero di incroci nel diagramma. Consideriamo i seguenti casi: #margin-note[ricontrollare tutta questa] + + - Se $K$ ha una sola componente ed un incrocio $i$. Consideriamo $S_i K$, $E_i K$ e $e_i K$, scegliendo bene $p$ possiamo fare in modo che $i$ sia il primo incrocio nella sequenza di scambi. A questo punto otteniamo la tesi calcolando la differenza nelle espansioni di $Omega_K (p)$ e $Omega_(S_i K) (p)$. + + - Se $K$ ha più di una componente e $i$ è un incrocio di una componente con se stessa. In questo caso la tesi segue per dalla @kauffman-poly-def utilizzando per induzione il caso ii.a). + + - Se $K$ ha più di una componente e $i$ è un incrocio tra due componenti diverse allora per gli addendi che non intersecano nessuna delle due componenti possiamo procedere per induzione. Altrimenti possiamo scegliere un punto base appropriato e procedere per induzione. +] + +#lemma[ + Sia $K, K'$ diagrammi di link equivalenti a meno di isotopia regolare. Allora $L_K = L_(K')$, ovvero $L_K$ è un invariante di isotopia regolare. +] + +#proof[ + Procediamo sempre per induzione e mostriamo l'invarianza per mosse che non aumentano il numero di incroci. + + - Mossa II: Ci sono più casi in base a se la mossa riguarda una o più componenti + + - Se la mossa è su una sola componente allora ci basta scegliere il punto base come segue + + #todo[disegnino] + + in questo modo i due incroci non compariranno nella sequenza di scambi. Tutti i termini di ii.a) saranno invarianti per mosse di tipo II e lo sarà anche $L_K$. + + - Se la mossa riguarda più componenti allora il caso peggiore è il seguente + + #todo[disegnino] + + Vediamo che $L[K] = L[S_2 S_1 K]$... + + - Mossa III: + + - Se la mossa è su fili di una sola componente allora allora possiamo scegliere il punto base sul filo che passa sopra, in questo modo avremo la situazione seguente: + + #todo[disegnino] + + In un caso possiamo procedere per induzione, se invece ci sono degli splice possiamo utilizzare le seguenti equivalenze di diagrammi e poi procedere per induzione + + #todo[disegnino] + + - Nel caso in cui la mossa riguardi fili di più componenti possiamo utilizzare l'identità della mossa precedente $L[K] = L[S_2 S_1 K]$ per semplificare la situazione e procedere per induzione. + + Questo completa la dimostrazione. ] \ No newline at end of file diff --git a/src/prelude.typ b/src/prelude.typ index 260cfdf..fb94e0d 100644 --- a/src/prelude.typ +++ b/src/prelude.typ @@ -131,3 +131,13 @@ )) #let scr(it) = text(features: ("ss01",), box($cal(it)$)) + +#let blank = { + $#h(0.125em) #{ + rect(width: 0.8em * 1.41, height: 0.8em, stroke: 0.5pt, place(dx: -0.2em, dy: -0.55em, { + set text(size: 10pt) + + $dots$ + })) + } #h(0.125em)$ +} \ No newline at end of file