Journal of Knot Theory and Its Ramifications

We construct a new invariant polynomial for long virtual knots. It is a generalization of the Alexander polynomial. We designate it as ζ by meaning an analogy with ζ-polynomial for virtual links. The degree of ζ-polynomial estimates the virtual crossing number. We describe some application of ζ-polynomial for the study of minimal long virtual diagrams with respect to the number of virtual crossings.

We introduce a new polynomial invariant of virtual knots and links and use this invariant to compute a lower bound on the virtual crossing number and the minimal surface genus.

We use quantum invariants to define a 3-manifold invariant j_p which lies in the non-negative integers. We relate j_p to the Heegaard genus, and the cut number. We show that j_p is an invariant of weak p-congruence.

This paper defines a new invariant of virtual knots and flat virtual knots. We study this invariant in two forms: the extended bracket invariant and the arrow polyomial. The extended bracket polynomial takes the form of a sum of virtual graphs with polynomial coefficients. The arrow polynomial is a polynomial with a finite number of variables for any given virtual knot or link. We show how the extended bracket polynomial can be used to detect non-classicality and to estimate virtual crossing number and genus for virtual knots and links.

Pairs of genus 2 mutant knots can have different Homfly polynomials, for example some 3-string satellites of Conway mutant pairs. We give examples which have different Kauffman 2-variable polynomials, answering a question raised by Dunfield et al. in their study of genus 2 mutants. While pairs of genus 2 mutant knots have the same Jones polynomial, given from the Homfly polynomial by setting v = s², we give examples whose Homfly polynomials differ when v = s³. We also give examples which differ in a Vassiliev invariant of degree 7, in contrast to satellites of Conway mutant knots.

We introduce a complexity c(X) ∈ ℕ for pairs X = (M,L), where M is a closed orientable 3-manifold and L ⊂ M is a link. The definition employs simple spines, but for well-behaved X's, we show that c(X) equals the minimal number of tetrahedra in a triangulation of M containing L in its 1-skeleton. Slightly adapting Matveev's recent theory of roots for graphs, we carefully analyze the behaviour of c under connected sum away from and along the link. We show in particular that c is almost always additive, describing in detail the circumstances under which it is not. To do so we introduce a certain (0, 2)-root for a pair X, we show that it is well-defined, and we prove that X has the same complexity as its (0, 2)-root. We then consider, for links in the 3-sphere, the relations of c with the crossing number and with the hyperbolic volume of the exterior, establishing various upper and lower bounds. We also specialize our analysis to certain infinite families of links, providing rather accurate asymptotic estimates.

The aim of this article is to detect new classes of quasi-alternating links. Quasi-alternating links are a natural generalization of alternating links. Their knot Floer and Khovanov homology are particularly easy to compute. Since knot Floer homology detects the genus of a knot as well as whether a knot is fibered, characterization of quasi-alternating links becomes an interesting open problem. We show that there exist classes of non-alternating Montesinos links, which are quasi-alternating.