Journal of Knot Theory and Its Ramifications

Generalized Reidemeister moves provide an extended set of moves to work with virtual knots and links. We introduce virtual tangle moves, generalization of classical rational tangle moves and show that such generalizations are essential to develop new invariants of virtual knots and links. We show that every 2-algebraic virtual link is a virtual 4-move equivalent to a trivial link or Hopf link. The properties of virtual tangle move are analyzed on few existing invariants associated with virtual knots and links.

The goal of this paper is to tabulate all genus one prime virtual knots having diagrams with ≤ 5 classical crossings. First, we construct all nonlocal prime knots in the thickened torus T × I which have diagrams with ≤ 5 crossings and admit no destabilizations. Then we use a generalized version of the Kauffman polynomial to prove that all those knots are different. Finally, we convert the knot diagrams in T thus obtained into virtual knot diagrams in the plane.

Manturov recently introduced the idea of a free knot, i.e. an equivalence class of virtual knots where equivalence is generated by crossing change and virtualization moves. He showed that if a free knot diagram is associated to a graph that is irreducibly odd, then it is minimal with respect to the number of classical crossings. Not all minimal diagrams of free knots are associated to irreducibly odd graphs, however. We introduce a family of free knot diagrams that arise from certain permutations that are minimal but not irreducibly odd.

Classical Hurwitz numbers of a fixed degree together with Hurwitz numbers of seamed surfaces give rise to a Klein topological field theory (see [A. Alexeevski and S. Natanzon, The algebra of bipartite graphs and Hurwitz numbers of seamed surfaces, Izv. Math. 72(4) (2008) 627–646]). We extend this construction to Hurwitz numbers of all degrees simultaneously. The corresponding infinite-dimensional Cardy–Frobenius algebra is computed in terms of Young diagrams and bipartite graphs. This algebra turns out to be isomorphic to the algebra of differential operators introduced in [A. Mironov, A. Morozov and S. Natanzon, Cardy–Frobenius extension of algebra of cut-and-join operators, J. Geom. Phys. 73 (2012) 243–251, arXiv:1210.6955; A Hurwitz theory avatar of open-closed string, Eur. Phys. J. C 73(2) (2013) 1–10, arXiv:1208.5057], which serves a model for open-closed string theory. We prove that the operators corresponding to Young diagrams and bipartite graphs give rise to relations between Hurwitz numbers.

We consider symmetries of spatial graphs in compact 3-manifolds described by smooth finite group actions. This paper provides a method for constructing an infinite family of hyperbolic spatial graphs with given symmetry by connecting spatial graph versions of hyperbolic tangles in 3-cells of polyhedral cell decompositions induced from triangulations of the 3-manifolds. This method is applicable also to the case of ideal triangulations.

We present a set of 26 finite quandles that distinguish (up to reversal and mirror image) by number of colorings, all of the 2977 prime oriented knots with up to 12 crossings. We also show that 1058 of these knots can be distinguished from their mirror images by the number of colorings by quandles from a certain set of 23 finite quandles. We study the colorings of these 2977 knots by all of the 431 connected quandles of order at most 35 found by Vendramin. Among other things, we collect information about quandles that have the same number of colorings for all of the 2977 knots. For example, we prove that if Q is a simple quandle of prime power order then Q and the dual quandle Q* of Q have the same number of colorings for all knots and conjecture that this holds for all Alexander quandles Q. We study a knot invariant based on a quandle homomorphism f : Q₁ → Q₀. We also apply the quandle colorings we have computed to obtain some new results for the bridge index, the Nakanishi index, the tunnel number, and the unknotting number. In an appendix we discuss various properties of the quandles in Vendramin's list. Links to the data computed and various programs in C, GAP and Maple are provided.

We give a diagrammatic definition of U_q(𝔰𝔩₂) when q is not a root of unity, including the Hopf algebra structure and relationship with the Temperley–Lieb category.

Bing, Bothe and Shilepsky studied knots that were wild at every point, and Bothe developed an invariant for a certain subclass of these knots. This paper develops invariants which distinguish ambient isotopy equivalence classes of these knots (which include the "Bing sling") and shows that there is an uncountable number of inequivalent of Bing sling type knots. As an aside, it is also shown that it is possible for inequivalent wild knots to have homeomorphic complements.

In this paper, we calculate the values of the E₆ state sum invariant for the lens spaces L(p, q). In particular, we show that the values of the invariant are determined by p mod 12 and q mod (p, 12). As a corollary, we show that the E₆ state sum is a homotopy invariant for the oriented lens spaces.