Index Polynomial Invariant of Virtual Links

Abstract

We introduce a polynomial invariant of virtual links that is non-trivial for many virtuals, but is trivial on classical links. Also this polynomial is sometimes useful to find the virtual crossing number of virtual knots. We give various properties of this polynomial and examples.

Crossing indices, traits and the principle of indistinguishability

“…Usually when an index is mentioned, one means the Z-valued index appeared in [3,28] or a Z-valued signed index introduced in [41,15,20,12]. In order to differ it from the general notion of index, we will call it the Gaussian index by analogy with the Gaussian parity [19].…”

“…The first appearance of an index can be found in V. Turaev's work [41] on strings (flat knots). After the paper of A. Henrich [15], the index appeared in works of several authors [3,20,28] and others. Z. Cheng [6] gave an axiomatic description of the index.…”

Some Polynomial Invariants of Welded Links

“…First we introduce an index polynomial for virtual links which is an extension of the polynomial invariant given in [4].…”

A note on coverings of virtual knots

“…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.…”