Tohoku Mathematical Journal
2017
December
SECOND SERIES VOL. 69, NO. 4

Tohoku Math. J.
69 (2017), 571-583

Title A POLYNOMIAL DEFINED BY THE $SL(2;\mathbb{C})$-REIDEMEISTER TORSION FOR A HOMOLOGY 3-SPHERE OBTAINED BY A DEHN SURGERY ALONG A $(2P,Q)$-TORUS KNOT

Author Teruaki Kitano

(Received June 4, 2015, revised November 12, 2015)
Abstract. Let $K$ be a $(2p,q)$-torus knot. Here $p$ and $q$ are coprime odd positive integers. Let $M_n$ be a 3-manifold obtained by a $1/n$-Dehn surgery along $K$. We consider a polynomial $\sigma_{(2p,q,n)}(t)$ whose zeros are the inverses of the Reidemeister torsion of $M_n$ for $\mathit{SL}(2;\mathbb{C})$-irreducible representations under some normalization. Johnson gave a formula for the case of the $(2,3)$-torus knot under another normalization. We generalize this formula for the case of $(2p,q)$-torus knots by using Tchebychev polynomials.

Mathematics Subject Classification. Primary 57M27.
Key words and phrases. Reidemeister torsion, torus knot, Brieskorn homology 3-sphere, $SL(2;\mathbb{C})$-representation.

To the top of this page

Back to the Contents