Topology Atlas | Conferences

Rotors and the homology of branched double covers of links and tangles
by
Jozef H. Przytycki
George Washington University
Coauthors: Jan Dymara (Ohio State University and Wroclaw University), Tadeusz Januszkiewicz (Wroclaw University)

There is a classical result that a mutation of a link, L, preserves the 2-fold branched cover of S³ with L as the branching set, M⁽²⁾_L. In particular, H₁(M⁽²⁾_L;Z) is preserved by a mutation. We address the analogous problem for a rotation of a link. More precisely, we ask for which n and p (p a prime number), n-rotation preserves H₁(M⁽²⁾_L);Z_p. We prove that the answer is positive if there is an s such that p^s = -1  mod(n) or if n=p. For example 4-rotation preserves H₁(M⁽²⁾_L);Z₂₁) but we have examples when a 4-rotation changes H₁(M⁽²⁾_L);Z₅). In our approach, we interpret H₁(M⁽²⁾_L);Z_p) using Fox p-coloring space, Col_p(L). We introduce a symplectic structure on the Z_p^2n-2 (space of boundary colorings), and analyze Z_n action on it. From the fact that tangles yield Lagrangians in Z_p^2n-2 we are able to gain information whether Lagrangians invariant under Z_p-action are also invariant under dihedral (D_n) action.

Date received: December 11, 2001

Copyright © 2001 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # caip-05.