On Gauss diagrams of periodic virtual knots

Abstract

In this paper, we study the Gauss diagrams for periodic virtual knots (Theorem 3.1) and show that the virtual knot corresponding to a periodic Gauss diagram is equivalent to the periodic virtual knot whose factor is the virtual knot corresponding to the factor Gauss diagram (Theorem 3.2). We give formulae for the writhe polynomial and the affine index polynomial of periodic virtual knots by using those of factor knots (Corollary 4.2, Corollary 4.6).

Writhe polynomial for virtual links

“…So, W Lc (t) can be seen as an index function. It automatically satisfies chord index axioms except (1). In fact, it does satisfy (1).…”

“…So, weight(c)(s) = 0 in this case. Hence, it satisfies (1). There is no difficulty to see that L c is still a chord index when we require c to be a self crossing point (Compare the proof of Theorem 3.9 in [7]).…”

“…For example, it is easy to compute and efficient to determine the virtuality of a large class of virtual knots. It also gives a constraint on periodicity of virtual knot (see [1]). Besides, it can be used to construct flat invariants easily (see [11]).…”

“…My strategy for constructing writhe polynomial of virtual links is to construct an new index function denoted by Ind ′ (c), which satisfys all chord axioms of Definition 3.1 except (1). There is indeed interrelationship between self crossing points and linking crossing points in Ind ′ (c) (see Example 5.12).…”

Alexander invariants of periodic virtual knots

“…So far, these basic questions have not been resolved. There is an assortment of known constraints on certain invariants of a virtual knot or link for it to be periodic, including conditions on the arrow and index polynomials [IL12], the Miyazawa polynomial [KLS09], the VA-polynomial [KLS13], the writhe and odd writhe polynomials [BL15], and the virtual Alexander polynomial [KLS14].…”