Index polynomials for virtual tangles

Abstract

We generalize the index polynomial invariant, originally introduced in [Tur08], [Hen10] and [Pet], to the case of virtual tangles. Three polynomial invariants result from this generalization; we give a brief overview of their definition and some basic properties. arXiv:1805.08178v1 [math.GT]

The multi-variable Affine Index Polynomial

“…While we are not surprised that the original AIP is a Vassiliev invariant of order one, as so are many other index-type polynomials, we believe to be the first to explicitly state and prove that the (original) Affine Index Polynomial is a Vassiliev invariant of order one. This is consistent with previous results [Pet18], where we showed that the Wriggle polynomial (which coincides with the AIP for virtual knots [FK13]) is an order one Vassiliev invariant.…”

“…The first nontrivial GPV invariant for closed knots happens for n = 3, while there are many interesting examples of Vassiliev invariants for n = 1, 2. Examples of Vassiliev invariants include the coefficients of a power series expansion of both the Conway polynomial and the Kauffman Bracket polynomial [Kau99], as well as various index polynomials ([Hen10], [Pet19], [Pet18]) and the Three Loop Isotopy invariant of [CD14].…”

Intersection formulas for parities on virtual knots

“…Kauffman [24], L.C. Folwaczny [11], J. Kim [27], M.J. Jeong [21], H. Gao, M. Xu [12], N. Petit [42], K. Kaur, M. Prabhakar, A. Vesnin [26] and others. The first axiomatization of index properties is due to Z. Cheng [5].…”