One conjecture on cut points of virtual links and the arrow polynomial of twisted links

TL;DR

This paper proves a conjecture about cut points in virtual links, extends the arrow polynomial to twisted links, and introduces new invariants for detecting checkerboard colorability in twisted links.

Contribution

It confirms a conjecture on cut points, generalizes the arrow polynomial to twisted links, and identifies characteristics for checkerboard colorability detection.

Findings

Proof of Dye's conjecture on cut points

Generalization of arrow polynomial to twisted links

Identification of characteristics for checkerboard colorability

Abstract

Checkerboard framings are an extension of checkerboard colorings for virtual links. According to checkerboard framings, in 2017, Dye obtained an independent invariant of virtual links: the cut point number. Checkerboard framings and cut points can be used as a tool to extend other classical invariants to virtual links. We prove that one of the conjectures in Dye's paper is correct. Moreover, we analyze the connection and difference between checkerboard framing obtained from virtual link diagram by adapting cut points and twisted link diagram obtained from virtual link diagram by introducing bars. By adjusting the normalized arrow polynomial of virtual links, we generalize it to twisted links. And we show that it is an invariant for twisted link. Finally, we figure out three characteristics of the normalized arrow polynomial of a checkerboard colorable twisted link, which is a tool of detecting checkerboard colorability of a twisted link. The latter two characteristics are the same as in the case of checkerboard colorable virtual link diagram.