Volume 8, issue 3 (2008)

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Twisted link theory

Mario O Bourgoin

Algebraic & Geometric Topology 8 (2008) 1249–1279
Abstract

We introduce stable equivalence classes of oriented links in orientable three-manifolds that are orientation I–bundles over closed but not necessarily orientable surfaces. We call these twisted virtual links and show that they subsume the virtual knots introduced by L Kauffman and the projective links introduced by Yu V Drobotukhina. We show that these links have unique minimal genus three-manifolds. We use link diagrams to define an extension of the Jones polynomial for these links and show that this polynomial fails to distinguish two-colorable links over nonorientable surfaces from non-two-colorable virtual links.

Keywords
virtual link, projective link, stable equivalence, Jones polynomial, fundamental group
Mathematical Subject Classification 2000
Primary: 57M25, 57M27, 57M15, 57M05
References
Publication
Received: 10 August 2006
Accepted: 14 November 2007
Published: 26 July 2008
Authors
Mario O Bourgoin
Department of Mathematics
Brandeis University
415 South Street, MS 050
Waltham, MA 02454
USA
http://people.brandeis.edu/~mob/