A Khovanov Type Invariant Derived From an Unoriented HQFT for Links in Thickened Surfaces

Abstract

Abstract. Two link diagrams on compact surfaces are strongly equivalent if they are related by Reidemeister moves and orientation preserving homeomorphisms of the surfaces. They are stably equivalent if they are related by the two previous operations and adding or removing handles. Turaev and Turner constructed a link homology for each stable equivalence class by applying an unoriented TQFT to a geometric chain complex similar to Bar-Natan's one. In this paper, by using an unoriented homotopy quantum field theory (HQFT), we construct a link homology for each strong equivalence class. Moreover, our homology yields an invariant of links (under ambient isotopy) in the oriented I-bundle of a compact surface.

Knot diagrams on a punctured sphere as a model of string figures

“…An affirmative answer above is first given by K. Tagami. His proof of Theorem 1.3 is based on his result in [3,Corollary 4.11]. After he told us his proof, we have noticed that there is a simple proof using Turaev cobracket [4] Theorem 1.4.…”