Journal of Knot Theory and Its Ramifications

Let P, Q be Heegaard surfaces of a closed orientable 3-manifold. In this paper, we introduce a method for giving an upper bound of (Hempel) distance of P by using the Reeb graph derived from a certain horizontal arc in the ambient space [0, 1] × [0, 1] of the Rubinstein–Scharlemann graphic derived from P and Q. This is a refinement of a part of Johnson's arguments used for determining stable genera required for flipping high distance Heegaard splittings.

In earlier work we introduced the graph bracket polynomial of graphs with marked vertices, motivated by the fact that the Kauffman bracket of a link diagram D is determined by a looped, marked version of the interlacement graph associated to a directed Euler system of the universe graph of D. Here we extend the graph bracket to graphs whose vertices may carry different kinds of marks, and we show how multiply marked graphs encode interlacement with respect to arbitrary (undirected) Euler systems. The extended machinery brings together the earlier version and the graph-links of Ilyutko and Manturov [J. Knot Theory Ramifications18 (2009) 791–823]. The greater flexibility of the extended bracket also allows for a recursive description much simpler than that of the earlier version.

We describe the link-symmetric groups of 2-bridge links by Schubert's normal forms and Conway's normal forms. The part of Schubert's normal forms is a collection of well-known facts. However, the part of Conway's normal forms may be new. Moreover, we deal with Conway's normal forms whose entries are even integers.

Self-avoiding polygons in the cubic lattice are models of ring polymers in dilute solution. The conformational entropy of a ring polymer is a dominant factor in its physical and chemical properties, and this is modeled by the large number of conformations of lattice polygons. Cubic lattice polygons are embeddings of the circle in three space and may be used as a model of knotting in ring polymers. In this paper we study the effects of knotting on the conformational entropy of lattice polygons and so determine the relative fraction of polygons of different knot types at large lengths. More precisely, we consider the number of cubic lattice polygons of n edges with knot type K, p_n(K). Numerical evidence strongly suggests that

A virtual string can be defined as an equivalence class of planar diagrams under certain kinds of diagrammatic moves. Virtual strings are related to virtual knots in that a simple operation on a virtual knot diagram produces a diagram for a virtual string. In this paper we consider three operations on a virtual string or virtual strings which produce another virtual string, namely covering, composition and cabling. In particular we study virtual strings unchanged by the covering operation. We also show how the based matrix of a composite virtual string is related to the based matrices of its components, correcting a result by Turaev. Finally we investigate what happens under cabling to some invariants defined by Turaev.