Journal of Knot Theory and Its Ramifications

Critical surfaces can be regarded as topological index 2 minimal surfaces which was introduced by David Bachman. In this paper, we give a sufficient condition and a necessary condition for self-amalgamated Heegaard surfaces to be critical.

In this paper, we investigate the behavior of the VA-polynomial over ℤ[t, t^-1] for periodic virtual knots and give a necessary condition for a given virtual knot to have period p. Indeed, we show that if K is a p-periodic virtual knot, then p divides the VA-polynomial of K. As an application, we improve the Im–Lee's table of possible periods for 117 virtual knots with classical crossings ≤4.

While Vassiliev invariants have proved to be a useful tool in the classification of knots, they are frequently defined through knot diagrams, and fail to illuminate any significant geometric properties the knots themselves may possess. Here, we provide a geometric interpretation of the second-order Vassiliev invariant by examining five-point cocircularities of knots, extending some of the results obtained in [R. Budney, J. Conant, K. P. Scannell and D. Sinha, New perspectives on self-linking, Adv. Math. 191(1) (2005) 78–113]. Additionally, an analysis on the behavior of other cocircularities on knots is given.

The mapping class group of a Heegaard splitting is the group of connected components in the set of automorphisms of the ambient manifold that map the Heegaard surface onto itself. We find examples of elements of the mapping class group that are periodic, reducible and pseudo-Anosov on the Heegaard surface, but are isotopy trivial in the ambient manifold. We prove structural theorems about the first two classes, in particular showing that if a periodic element is trivial in the mapping class group of the ambient manifold, then the manifold is not hyperbolic.

Region crossing change for a knot or a proper link is an unknotting operation. In this paper, we provide a sharp upper bound on the region unknotting number for a large class of torus knots and proper links. Also, we discuss conditions on torus links to be proper.

If the exterior E(G) of a spatial graph G in a closed orientable 3-manifold is an irreducible 3-manifold with incompressible boundary, there is a unique finite graph Γ associated to the JSJ decomposition of E(G). This paper provides a method for constructing spatial graphs such that a given finite connected graph is associated to the JSJ decompositions of their exteriors.

We describe a method of encoding various types of link diagrams, including those with classical, flat, rigid, welded, and virtual crossings. We show that this method may be used to encode link diagrams, up to equivalence, in a notation whose length is a cubic function of the number of "riser marks". For classical knots, the minimal number of such marks is twice the bridge index, and a classical knot diagram in minimal bridge form with bridge index b may be encoded in space . A set of moves on the notation is defined. As a demonstration of the utility of the notation we give another proof that the Kishino virtual knot is non-classical.