From 0a13243240fcfda27d34fc329a7868e258cf662b Mon Sep 17 00:00:00 2001 From: Antonio De Lucreziis Date: Sun, 6 Jul 2025 12:45:51 +0200 Subject: [PATCH] fixed a proof --- out/tesi-triennale.pdf | Bin 1706356 -> 1709017 bytes src/main.typ | 54 ++++++++++++++++++++++++++++++++--------- src/prelude.typ | 4 ++- src/theme.typ | 6 +++-- 4 files changed, 49 insertions(+), 15 deletions(-) diff --git a/out/tesi-triennale.pdf b/out/tesi-triennale.pdf index f861d94f981976875669425c2e156096e1e39645..234bda7b52dd3d6b13971228b310c20eec9f721d 100644 GIT binary patch delta 33404 zcmaHS1yEeg5-y7bcUv5S6Ck*|yE_4bdvKQpLV)1DgamhYHo=2SaQEQuPJp~k^5_1q z>fZOLI#pfM)6@O+^vvw}=5U#4GhbwbL8~Su!@|zWi$WU&{Yz+PNUCVSO^{b7=VGH| zqtq}$p`?7N<)r<_*4)X$l9G$#S0jpuNYW$U^CY!K`UH8RBsG5cByxZqfIaD{k)o~` z&;^%tfrMWdgy04TuE_-A*C8SQe3@jAfC0t~C3==bIZB$u%U=xOOiC2s`Q7*qp2SHH0&f63P*+5Cp@O`yNl9Y(bz{O^umC>r8aysuU9ngKRZ_Ds zM$*bGN>X(_Fv-@e2EdmzWd6q&i3Kpp-<*$}`+vdX{tvv^`acdyEW)6;-i(vgwOe#N 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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) @@ -806,21 +808,21 @@ sempre con $delta = (a + a^(-1)) slash z - 1$. 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))) + ((-1)^(abs(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 + 2. #marker([iii.b)]) 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 + L_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)^(abs(lambda(q))+1) kL(hat(K)(lambda(q))) + z sum_K (lambda(q))) ] $ ] @@ -828,6 +830,8 @@ sempre con $delta = (a + a^(-1)) slash z - 1$. == Dimostrazione buona definizione +Bla bla @kauffman-rec-single-component. + Per prima cosa osserviamo che dato che in ogni termine della definizione induttiva per $L_K$, quando questa dipende dalla scelta di un punto base sono presenti i termini per entrambe le direzioni; quindi ci basta dimostrare induttivamente che le definizioni non dipendono dalla scelta di punto base. Nel corso della dimostrazione tutti gli argomenti per induzione si baseranno sul numero di incroci dei diagrammi di link. Ometteremo i casi base ovvero i controlli per diagrammi con pochi vertici che fanno partire l'induzione in quanto sono tutti facili da controllare. @@ -1080,15 +1084,35 @@ $ ] #lemma[ - Sia $K$ un diagramma di un nodo. Allora ii.b) non dipende dalla scelta di punto base e più precisamente vale + Sia $K$ un diagramma di un nodo. Allora @kauffman-rec-single-component non dipende dalla scelta di punto base. Inoltre se $p$ è un punto base direzionato su $K$ e $lambda(p)$ la sequenza di scambi determinata da $p$ allora $ - Omega_K(p) = (-1)^(abs(lambda(p)) + 1) L_hat(K)(p) + z sum_K (lambda(p)) + Omega_K (p) = (-1)^(abs(lambda(p)) + 1) L_(hat(K)(p)) + z sum_K (lambda(p)) $ + + non dipende dalla scelta di punto base. ] #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 $p$. + Possiamo assumere che la sequenza di scambi determinata da $p$ sia etichetta $lambda = (n, dots, 0)$, in questo modo + + $ + Omega_K (p) = (-1)^(n+1) L_(hat(K)(p)) + z sum_K (lambda(p)) + $ + + Ricordiamo come era definito $L_K$ nel caso di una sola componente @kauffman-rec-single-component e notiamo quanto segue + + $ + L_K (a, z) &colon.eq 1 / 2 [ + sum_(q = p, overline(p)) + ((-1)^(abs(lambda(q))+1) kL(hat(K)(lambda(q))) + z sum_K (lambda(q))) + ] \ + &= 1/2 (Omega_K (p) + Omega_K (overline(p))) + $ + + Dunque ci basta mostrare che $Omega_K (p)$ non dipenda dalla scelta di punto base. + + Come in precedenza possiamo mostrarlo facendo scorrere $p$ nell'arco successivo al primo incrocio dopo $p$ (questo ci induce l'invarianza per permutazioni cicliche e ci permette di concludere). 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: @@ -1181,13 +1205,19 @@ $ ] #proof[ - Mostriamo per induzione sul numero di incroci nel diagramma. Consideriamo i seguenti casi: #margin-note[ricontrollare tutta questa] + Mostriamo per induzione sul numero di incroci nel diagramma. + + 1. 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 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 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. - - 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). + 2. La dimostrazione è analoga, basta osservare che possiamo sempre scegliere un punto base che non faccia comparire l'incrocio del ricciolo nella sequenza di scambi. - - 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. + E questo conclude la dimostrazione del lemma. ] #lemma[ diff --git a/src/prelude.typ b/src/prelude.typ index fb94e0d..9265423 100644 --- a/src/prelude.typ +++ b/src/prelude.typ @@ -140,4 +140,6 @@ $dots$ })) } #h(0.125em)$ -} \ No newline at end of file +} + +#let marker(content) = metadata(("marker", content)) \ No newline at end of file diff --git a/src/theme.typ b/src/theme.typ index 6924287..4d26bc4 100644 --- a/src/theme.typ +++ b/src/theme.typ @@ -308,7 +308,7 @@ show ref: it => { if it.element != none { let el = it.element - if el.kind == "definition" or el.kind == "proposition" or el.kind == "lemma" or el.kind == "theorem" { + if el.func() == figure and (el.kind == "definition" or el.kind == "proposition" or el.kind == "lemma" or el.kind == "theorem") { link(el.location(), { if it.supplement != auto { it.supplement } else { el.supplement } [ ] @@ -317,8 +317,10 @@ numbering(el.numbering, ..el.counter.at(el.location())), ).join(".") }) + } else if el.func() == metadata and type(el.value) == array and el.value.at(0) == "marker" { + link(el.location(), el.value.at(1)) } else { - repr(el) + repr(it) } } else if it.citation != none { it