some inkspace pictures

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lectures_on_knot_theory.tex

@@ -550,8 +550,12 @@ If $K$ is a knot, then $n$ is necessarily even, and so $\Delta_K(t^{-1}) = t^{-n
 \end{align*}
 \end{lemma}
 \begin{proof}
-
+If $\Sigma$ is a genus $g$ - Seifert surface for $K$ then $H_1(\Sigma) = \mathbb{Z}^{2g}$, so $S$ is an $2g \times 2g$ matrix. Therefore $\det (tS - S^T)$ is a polynomial of degree at most $2g$.
 \end{proof}
+\begin{example}
+There are not trivial knots with Alexander polynomial equal $1$, for example: 
+$\Delta_{11n34} \equiv 1$. 
+\end{example}
 %removing one disk from surface doesn't change $H_1$ (only $H_2$)
 \section{}
 \begin{example}