Title : Reidemeister torsion and lens surgeries on $(-2, m, n)$-pretzel knots
(a joint work with Yuichi Yamada (The University of Electro-Communications))
Speaker : Teruhisa Kadokami (Osaka City University)
Abstract :

We define the {\it Reidemeister torsion} following V.~G.~Turaev. Any homology lens space is obtained by $p/q$-surgery along a knot in a homology 3-sphere $\Sigma$, where $|p|\ge 2$ and $q\ne 0$. We denote it by $\Sigma (K;p/q)$. We call a homology lens space $\Sigma (K;p/q)$ is {\it of lens type} if its Reidemeister torsions are equal to those of a lens space. Let $P(-2, m, n)$ be the $(-2, m, n)$-pretzel knot in $S^3$, where $m$ and $n$ are odd numbers with $m, n\ge 3$, and ${\varDelta}_{m, n}(t)$ the Alexander polynomial of $P(-2, m, n)$. We study a knot in a homology 3-sphere whose Alexander polynomial is ${\varDelta}_{m, n}(t)$ by using Reidemeister torsion. Then we obtained the following three theorems: \medskip {\noindent {\bf Theorem 1}}\ {\it Let $K$ be a knot in a homology 3-sphere $\Sigma$ whose Alexander polynomial is ${\varDelta}_{m, n}(t)$. Then} \medskip {\noindent (1)}\ {\it If $\Sigma(K;p/q)$ is of lens type and $p$ is divisible by $2$, then $\{m, n\}=\{3, 5\}$ or $\{3, 7\}$.} \medskip {\noindent (2)}\ {\it If $\Sigma(K;p/q)$ is of lens type and $p$ is divisible by $4$, then $\{m, n\}=\{3, 5\}$.} \medskip {\noindent {\bf Theorem 2}}\ {\it Let $K$ be a knot in a homology 3-sphere $\Sigma$ whose Alexander polynomial is ${\varDelta}_{3, n}(t)$\ {\rm ({\it $n\ge 7$ and $n$ is odd})}. Then} \medskip {\noindent (1)}\ {\it If $\Sigma(K;(2n+4)/q)$ is of lens type, then $n=7$ and $q\equiv \pm 1\ (\mathrm{mod}\ \! 2n+4)$.} \medskip {\noindent (2)}\ {\it If $\Sigma(K;(2n+5)/q)$ is of lens type, then $n=7$ and $q\equiv \pm 1\ (\mathrm{mod}\ \! 2n+5)$.} \medskip {\noindent {\bf Theorem 3}}\ {\it If $\Sigma(K;p/q)$ is of lens type, then the Reidemeister torsion of $\Sigma(K;p/q)$ is the same as that of $L(p, qi^2)$.}

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