The Forbidden Quiver of a Link

Abstract

The forbidden moves in virtual knot theory can be used to unknot any knot, virtual or classical; however, multi-component crossings in links can still survive, resulting a fused link. Using the forbidden moves, we categorify fused links obtain a quiver-valued invariant of classical and virtual links we call the forbidden quiver, opening the way for functors to and from other categories. As an application we use the forbidden quiver to obtain three polynomial invariants of virtual and classical links. Since these invariants are not sensitive to single-component crossing change, they are also link homotopy invariants.

10 pages. Last version includes corrected diagrams for forbidden moves, which are reversed in the published version