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transform="matrix(0.38109416,0,0,0.32789967,-45.883691,-38.363019)"> - + ry="125.81031" + rx="127.15614" + cy="663.07581" + cx="310.73593" + id="path9919" + style="opacity:1;fill:#0000ff;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:1.21091163;stroke-linejoin:bevel;stroke-miterlimit:4;stroke-dasharray:1.21091165, 1.21091165;stroke-dashoffset:0;stroke-opacity:1" /> + + + \shortstack{genus $0$}  genus $2$  \shortstack{genus $1$}  genus $3$  + d="M 620.6719,60.42292 C 615.59149,94.42406 549.976,88.6098 521.84095,73.36683 506.80675,80.70638 479.49958,88.51649 459.52671,80.73227 379.9298,48.24636 484.01556,4.64937 525.75154,38.74941 549.6689,28.76098 621.75431,20.77284 620.6719,60.42292 Z" + style="opacity:1;fill:#0000ff;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:0.93750006;stroke-linejoin:bevel;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1" + inkscape:connector-curvature="0" /> - - - - genus $0$  genus $2$  genus $1$  genus 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+\includegraphics[width=0.6\textwidth]{seifert_connect.png} \end{center} \caption{Connecting two surfaces.} \label{fig:SeifertConnect} @@ -239,11 +239,11 @@ Note: in general the obtained surface doesn't need to be connected, but by takin Every link in $S^3$ bounds a surface $\Sigma$ that is compact, connected and orientable. Such a surface is called a Seifert surface. \end{theorem} % -\begin{figure}[H] -\fontsize{15}{10}\selectfont +\begin{figure}[h] +\fontsize{12}{10}\selectfont \centering{ \def\svgwidth{\linewidth} -\resizebox{0.8\textwidth}{!}{\input{images/torus_1_2_3.pdf_tex}} +\resizebox{1\textwidth}{!}{\input{images/torus_1_2_3.pdf_tex}} \caption{Genus of an orientable surface.} \label{fig:genera} } @@ -277,7 +277,7 @@ Let $\nu(\beta)$ be a tubular neighbourhood of $\beta$. The linking number can \begin{itemize} \item Hopf link -\begin{figure}[H] +\begin{figure}[h] \fontsize{20}{10}\selectfont \centering{ \def\svgwidth{\linewidth} @@ -286,7 +286,7 @@ Hopf link \end{figure} \item $T(6, 2)$ link -\begin{figure}[H] +\begin{figure}[h] \fontsize{20}{10}\selectfont \centering{ \def\svgwidth{\linewidth} @@ -686,12 +686,15 @@ Let $X$ be the four-manifold obtained via the double branched cover of $B^4$ bra \end{fact} \noindent Let $Y = \Sigma(K)$. Then: -\begin{align*} -&H_1(Y, \mathbb{Z}) \times H_1(Y, \mathbb{Z}) \longrightarrow \quot{\mathbb{Q}}{\mathbb{Z}}\\ &(a,b) \mapsto a A^{-1} b^{T},\qquad -A = V + V^T\\ -&H_1(Y, \mathbb{Z}) \cong \quot{\mathbb{Z}^n}{A\mathbb{Z}}\\ -&A \longrightarrow BAC^T \quad \text{Smith normal form} -\end{align*} +\begin{flalign*} +H_1(Y, \mathbb{Z}) \times H_1(Y, \mathbb{Z}) \longrightarrow \quot{\mathbb{Q}}{\mathbb{Z}}& +\\ +(a,b) \mapsto a A^{-1} b^{T},\qquad +A = V + V^T& +\\ +H_1(Y, \mathbb{Z}) \cong \quot{\mathbb{Z}^n}{A\mathbb{Z}}&\\ +A \longrightarrow BAC^T \quad \text{Smith normal form}& +\end{flalign*} ???????????????????????\\ In general @@ -762,23 +765,23 @@ H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) &\longrightarrow \quot{\mathbb{Q}}{\ma \end{fact} \noindent Note that $\mathbb{Z}$ is not PID. Therefore we don't have primer decomposition of this moduli. We can simplify this problem by replacing $\mathbb{Z}$ by $\mathbb{R}$. We lose some date by doing this transition. -\begin{align*} -&\xi \in S^1 \setminus \{ \pm 1\} +\begin{flalign*} +\xi \in S^1 \setminus \{ \pm 1\} \quad p_{\xi} = -(t - \xi)(1 - \xi^{-1}) t^{-1}\\ -&\xi \in \mathbb{R} \setminus \{ \pm 1\} +(t - \xi)(1 - \xi^{-1}) t^{-1}&\\ +\xi \in \mathbb{R} \setminus \{ \pm 1\} \quad -q_{\xi} = (t - \xi)(1 - \xi^{-1}) t^{-1}\\ -&\xi \notin \mathbb{R} \cup S^1 \quad -q_{\xi} = (t - \xi)(t - \overbar{\xi})(1 - \xi^{-1})(1 - \overbar{\xi}^{-1}) t^{-2}\\ -&\Lambda = \mathbb{R}[t, t^{-1}]\\ -&\text{Then: } H_1(\widetilde{X}, \Lambda) \cong \bigoplus_{\substack{\xi \in S^1 \setminus \{\pm 1 \}\\ k\geq 0}} +q_{\xi} = (t - \xi)(1 - \xi^{-1}) t^{-1}&\\ +\xi \notin \mathbb{R} \cup S^1 \quad +q_{\xi} = (t - \xi)(t - \overbar{\xi})(1 - \xi^{-1})(1 - \overbar{\xi}^{-1}) t^{-2}&\\ +\Lambda = \mathbb{R}[t, t^{-1}]&\\ +\text{Then: } H_1(\widetilde{X}, \Lambda) \cong \bigoplus_{\substack{\xi \in S^1 \setminus \{\pm 1 \}\\ k\geq 0}} ( \quot{\Lambda}{p_{\xi}^k })^{n_k, \xi} \oplus \bigoplus_{\substack{\xi \notin S^1 \\ l\geq 0}} -(\quot{\Lambda}{q_{\xi}^l})^{n_l, \xi} -\end{align*} +(\quot{\Lambda}{q_{\xi}^l})^{n_l, \xi}& +\end{flalign*} We can make this composition orthogonal with respect to the Blanchfield paring. \vspace{0.5cm}\\ Historical remark: @@ -893,10 +896,10 @@ Suppose $A$ and $B$ are two symmetric polynomials that are coprime and that $\fo \begin{proof}[Idea of proof] For any $z$ find an interval $(a_z, b_z)$ such that if $P(z) \in (a_z, b_z)$ and $P(z)A(z) + Q(z)B(z) = 1$, then $Q(z) > 0$, $x(z) = \frac{az + bz}{i}$ is a continues function on $S^1$ approximating $z$ by a polynomial . \\??????????????????????????\\ -\begin{align*} -(1, 1) \mapsto \frac{h}{p^k} \mapsto \frac{g\overbar{g}h}{p^k}\\ -g\overbar{g} h + p^k\omega = 1 -\end{align*} +\begin{flalign*} +(1, 1) \mapsto \frac{h}{p^k} \mapsto \frac{g\overbar{g}h}{p^k}&\\ +g\overbar{g} h + p^k\omega = 1& +\end{flalign*} Apply Lemma \ref{L:coprime polynomials} for $A=h$, $B=p^{2k}$. Then, if the assumptions are satisfied, \begin{align*} Ph + Qp^{2k} = 1\\ @@ -927,7 +930,7 @@ If $P$ has no roots on $S^1$ then $B(z) > 0$ for all $z$, so the assumptions of \begin{theorem}(Matumoto, Conway-Borodzik-Politarczyk) Let $K$ be a knot, \begin{align*} -H_1(\widetilde{X}, \Lambda) \times +&H_1(\widetilde{X}, \Lambda) \times H_1(\widetilde{X}, \Lambda) = \bigoplus_{\substack{k, \xi, \epsilon\\ \xi in S^1}} (\quot{\Lambda}{p_{\xi}^k}, \epsilon)^{n_k, \xi, \epsilon} \oplus \bigoplus_{k, \eta}