Quaternionic Invariants of Virtual Knots and Links

Abstract

In this paper, we define and give examples of a family of polynomial invariants of virtual knots and links. They arise by considering certain 2 × 2 matrices with entries in a possibly non-commutative ring, for example, the quaternions. These polynomials are sufficiently powerful to distinguish the Kishino knot from any classical knot, including the unknot.

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“…The non-trivial knot depicted in the figure is obtained as the connected sum of two trivial knots. The non-triviality of the Kishino knot can be proved in various ways: by considering the Kauffman polynomial of the structure of the knot (see [18]), using Fenn's approach using quaternionic bigroups ( [1]) and considering the polynomial Ξ, which is a modification of the Jones-Kauffman polynomial put forward in [22]. In particular, it follows from the non-triviality of the Kishino knot that each of the two long knots forming it is non-trivial.…”

“…For the generalized moves one considers Reidemeister virtual moves consisting of Reidemeister classical moves and also virtual moves that can be regarded as versions of a single "bypass" move; more precisely, the branch of the knot containing only interior crossings can be removed and resurrected anywhere else in the plane (with the same endpoints). A more detailed description of the theory of virtual knots can be found in [5,15,20,21,26,36,37,38]. Virtual knots can also be interpreted topologically as knots in "thickened surfaces" S g ×I; the latter is the Cartesian product of a sphere with handles S g and the closed interval I, and the virtual knots are regarded to within isotopy and stabilization of these surfaces, that is, gluing and removal of new "thickened" handles which have empty intersection with the knot under consideration.…”

“…Virtual knots can also be interpreted topologically as knots in "thickened surfaces" S g ×I; the latter is the Cartesian product of a sphere with handles S g and the closed interval I, and the virtual knots are regarded to within isotopy and stabilization of these surfaces, that is, gluing and removal of new "thickened" handles which have empty intersection with the knot under consideration. In [19] Kuperberg showed that every virtual knot has a canonical minimal representation in S g × I with minimal g. 1 A corollary of Kuperberg's theorem is that the classical theory of knots can be embedded in the theory of virtual knots in the sense that two classical virtual knot diagrams are virtually equivalent if and only if they give isotopic knots in the usual sense. The first proof of this fact was given in [7].…”

“…Furthermore, long virtual knots do not commute; that is, the connected sum of two long knots K 1 #K 2 is not in general equivalent to the connected sum K 2 #K 1 , and so on. In the papers [20,21,22,25,26,35,36,37] and in many other papers, numerous invariants of virtual knots have been constructed.…”

On Virtual Crossing Number Estimates for Virtual Links

“…We attack the virtual crossing number by using the ξ-polynomial introduced independently by several authors (see [11,13,18,19], for the proof of their coincidence, see [2], for detecting non-classicality see also [20]). Sometimes it is called a generalized Alexander polynomial and denoted by ∆ 0 (see [4]).…”