New York Journal of Mathematics
Volume 28 (2022), 1372-1398


Hans U. Boden and Homayun Karimi

Classical results for alternating virtual links

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Published: October 9, 2022.
Keywords: Alternating link, virtual link, split link, checkerboard coloring, determinant, almost classical link, Alexander polynomial, welded link, double branched cover, Tait conjectures.
Subject [2010]: Primary: 57K10, 57K12.

We extend some classical results of Bankwitz, Crowell, and Murasugi to the setting of virtual links. For instance, we show that an alternating virtual link is split if and only if it is visibly split, and that the Alexander polynomial of any almost classical alternating virtual link is alternating. The first result is a consequence of an inequality relating the link determinant and crossing number for any non-split alternating virtual link. The second is a consequence of the matrix-tree theorem of Bott and Mayberry. We extend the first result to semi-alternating virtual links. We discuss the Tait conjectures for virtual and welded links and note that Tait's second conjecture is not true for alternating welded links.


This paper is based on several ideas in the Ph.D. thesis of the second author. The authors would like to thank Robin Gaudreau, Andy Nicas, Will Rushworth, and Adam Sikora for their helpful comments and feedback. They would also like to thank the referee for their input. The first author gratefully acknowledges grant funding from the Natural Sciences and Engineering Research Council of Canada.

Author information

Hans U. Boden:
Mathematics & Statistics
McMaster University
Hamilton, Ontario, Canada


Homayun Karimi:
Mathematics & Statistics
McMaster University
Hamilton, Ontario, Canada