Fundamental group of the double branch cover of S<sup>3</sup> along 2-bridge knots

Mieczyslaw K. Dabkowski

Knots in Washington XVII, Conference on Knot Theory and its Ramifications
December 19-21, 2003
GWU
Washington, DC, USA

Organizers
Marta M.Asaeda (U.Iowa), Mietek K.Dabkowski (UTD), Jozef H.Przytycki(GWU), Yongwu Rong (GWU)

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Fundamental group of the double branch cover of S³ along 2-bridge knots
by
Mieczyslaw K. Dabkowski
University of Texas at Dallas
Coauthors: Jozef H. Przytycki (GWU), Amir A. Togha (GWU)

The associated core group of a link diagram, \Pi(D⁽²⁾ was introduced by R.Fenn and C.Rourke (following the core group of Joyce).

\Pi(D⁽²⁾ is the group associated to the diagram D as follows: generators of G_D correspond to arcs of the diagram. Any crossing v_s yields the relation r_s=y_iy_j^-1y_iy_k^-1 where y_i corresponds to the overcrossing and y_j, y_k correspond to the undercrossings at v_s.

The topological interpretation of G_D was given by M.Wada: \Pi(D⁽²⁾ is the free product of the fundamental group of the cyclic branched double cover of S³ with branching set L and the infinite cyclic group. That is: \Pi(D⁽²⁾=\pi₁((M_L)⁽²⁾) * Z.

We give a very simple proof of the Wada theorem using Wirtinger presentation of the fundamental group of a link. We show how to use our construction to find (well organized) presentations of fundamental groups of double branched cover along 2-bridge knots.

In the follow up talk by A.Thoga the construction will be use to show that some of our groups are not left-orderable.

Date received: December 10, 2003