On arrow polynomials of checkerboard colorable virtual links

Abstract

In this paper we give two new criteria of detecting the checkerboard colorability of virtual links by using odd writhe and arrow polynomial of virtual links, respectively. By applying new criteria, we prove that 6 virtual knots are not checkerboard colorable, leaving only one virtual knot whose checkerboard colorability is unknown among all virtual knots up to four classical crossings.

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

“…Hence, the remain proof is consistent with the virtual link diagram case in [5] (see the proof of Theorem 4.3(1) and ( 2)).…”

“…Note that we do not read out the labeling b of bars, that is, there is not b in the word since the information of bars is transferred to the word. Note that the word is consistent with the virtual link diagram case in [5] (see the proof of Theorem 4.3).…”

“…In this section, we recall arrow polynomial of an oriented virtual link diagram defined by Dye and Kauffman [6] which is a criterion of detecting checkerboard colorability of virtual links in [5].…”

“…Next we shall explain that the Theorem 3.12(2) and 3.12(3) for the case of twisted link diagram can be reduced to the case of virtual link diagram in [5] (Theorem 4.3(1) and ( 2 Let σ be an unreduced state of B without applying reduction rule in Fig. 16.…”