2017/06/23 by Micah Chrisman, Chrisman, Micah, Robert G. Todd +1
Computer Science · Mathematics · #57M25 #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Logic, programming, and type systems
paper · pdf · doi:10.48550/arxiv.1706.07756
openalex publication_date 2017/06/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Geometric interpretations of some virtual knot invariants are given in terms of invariants of links in \mathbbS3. Alexander polynomials of almost classical knots are shown to be specializations of the multi-variable Alexander polynomial of certain two-component boundary links of the form J \sqcup K with J a fibered knot. The index of a crossing, a common ingredient in the construction of virtual knot invariants, is related to the Milnor triple linking number of certain three-component links J \sqcup K1 \sqcup K2 with J a connected sum of trefoils or figure-eights. Our main technical tool is virtual covers. This technique, due to Manturov and the first author, associates a virtual knot υ to a link J \sqcup K, where J is fibered and lk(J,K)=0. Here we extend virtual covers to all multicomponent links L=J \sqcup K, with K a knot. It is shown that an unknotted component J0 can be added to L so that J0 \sqcup J is fibered and K has algebraic intersection number zero with a fiber of J0 \sqcup J. This is called fiber stabilization. It provides an avenue for studying all links with virtual knots.