Twisted Link Theory

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 links, and show that they subsume the virtual knots introduced by L. Kauffman, and the projective links introduced by Yu. 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 non-orientable surfaces from non-two-colorable virtual links.

33 pages and 35 figures

Cited by in corpus (11)

• Turaev Hyperbolicity of Classical and Virtual Knots
• Additional Gradings in Khovanov Homology
• Prime Decomposition and Non-Commutativity in the Monoid of Long Virtual Knots
• Biquandles with structures related to virtual links and twisted links
• Double coverings of twisted links
• Surface pole bracket polynomials of virtual knots and twisted knots
• Twisted virtual biracks and their twisted virtual link invariants
• Virtual links which are equivalent as twisted links
• An analog of the {K}auffman bracket polynomial for knots in the non-orientable thickening of a non-orientable surface
• Links in the spherical 3-manifold obtained from the quaternion group and their lifts
• One conjecture on cut points of virtual links and the arrow polynomial of twisted links