New York Journal of Mathematics
Volume 24 (2018), 630-660


Micah Chrisman and Robert G. Todd

Relating virtual knot invariants to links in the 3-sphere

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Published: September 3, 2018
Keywords: virtual knots, virtual covers, multi-variable Alexander polynomial, boundary links, index polynomial, Milnor triple-linking number.
Subject: Primary: 57M25, Secondary: 57M27.

Geometric interpretations of some virtual knot invariants are given in terms of invariants of links in S3. 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 ⊔ 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 ⊔ K_1 ⊔ K_2 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 v to a link J ⊔ K, where J is fibered and lk(J,K)=0. Here we extend virtual covers to all multi-component links L=J ⊔ K, with K a knot. It is shown that an unknotted component J0 can be added to L so that J0 ⊔ J is fibered and K has algebraic intersection number zero with a fiber of J0 ⊔ J. This is called fiber stabilization. It provides an avenue for studying all links with virtual knots.


The first author acknowledges a Spring 2015 Creativity and Research Grant from Monmouth University. These funds allowed him to visit the second author at the University of Nebraska-Omaha and to complete Section 3. Both authors are grateful for helpful discussions and encouragement from C. Frohman and H. U. Boden. Lastly we would like to thank an anonymous referee for a close reading of a previous draft of this paper that led to an improvement in the statement of Theorem 2.1.1. The referee also improved the paper by pointing out some errors in the original arXiv version (arXiv:1706.07756v1[math.GT], Section 2.2) of this paper.

Author information

Micah Chrisman:
Department of Mathematics, The Ohio State University, Marion, Ohio, USA

Robert G. Todd:
Department of Natural and Applied Sciences, Mount Mercy University, Cedar Rapids, Iowa, USA