The writhes of a virtual knot

Abstract

Kauffman introduced a fundamental invariant of a virtual knot called the odd writhe. There are several generalizations of the odd writhe, such as the index polynomial and the odd writhe polynomial. In this paper, we define the n-writhe for each non-zero integer n, which unifies these invariants, and study various properties of the n-writhe.

Writhe polynomials and shell moves for virtual knots and links

“…For each integer n, we denote by J n (G) the sum of signs of all chords γ with Ind(γ) = n. If n = 0, then J n (G) does not depend on a particular choice of G of K [12]. It is called the n-writhe of K and denoted by J n (K).…”

A note on coverings of virtual knots

“…The index of c is the sum of signs of endpoints of chords on α, and denoted by Ind G (c) (cf. [1,9,11]). In the case that G is linear, the index of c is defined as that of c in the closure of G. Let G be a circular or linear Gauss diagram.…”

“…Let G be a Gauss diagram of a (long) virtual knot K. For an integer n = 0, we denote by w n (G) the sum of signs of all chords c of G with Ind G (c) = n. Then w n (G) does not depend on a particular choice of G of K; that is, w n (G) is an invariant of K. In [11], the proof is given for a virtual knot, and the case of a long virtual knot is similarly proved. It is called the n-writhe of K and denoted by w n (K).…”

“…Later the notion of index is introduced in [2,4,6,11] which assigns an integer to each real crossing such that the parity of the index coincides with the original parity. The n-writhe is defined as a refinement of the odd writhe.…”

On knots with no 3-state

“…The third author and K. Taniguchi [5] showed that the n-writhe of D is an invariant of a virtual knot if n = 0. The n-writhe J n (K) of K is defined by J n (K) = J n (D).…”