The Forbidden Quiver of a Link
Sam Nelson, Stella Shah
TL;DR
The paper addresses classification of fused links under forbidden moves in virtual knot theory by introducing a finite signed quiver invariant called the forbidden quiver $FQ(L)$. The construction yields a categorification since quivers form small categories and leads to decategorified invariants $FM(L)$, its determinant, and polynomials $Phi_F(L)$, $Phi^2_F(L)$, and $Phi_F^{MP}(L)$. It proves invariance under Reidemeister moves and link homotopy, notes that single-component crossing changes are undetected, and computes several examples, including classical knots up to seven crossings. The work bridges Gauss-diagram techniques, quiver theory, and categorification, and opens avenues for functorial interpretations to other categories and for studying fused links and virtual links.
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.