Journal of Knot Theory and Its Ramifications

In this paper, we discuss the region unknotting number of different classes of 2-bridge knots. In particular, we provide region unknotting number for the classes of 2-bridge knots whose Conway notations are C(m, n), C(m, 2, m), C(m, 2, m ± 1) and C(2, m, 2, n). By generalizing, we also provide a sharp upper bound for all the remaining classes of 2-bridge knots.

We study the properties of a knot K to be 2-adjacent to another knot W by analyzing their Conway polynomials, Jones polynomials and Homfly polynomials and give some very useful conditions. We discuss whether each pair of knots can be 2-adjacent to each other, i.e. whether 2-adjacency is a symmetric relation. We discuss also whether the trivial knot, the trefoil knot and the figure-eight knot can be 2-adjacent to any knot in Rolfsen's table and the opposite cases, except for 9₃₄ it is not decided whether it is 2-adjacent to 4₁. Finally, we give some examples to answer I. Torisu's problem partly and etc.

By methods similar to those used by L. Jeffrey [L. C. Jeffrey, Chern–Simons–Witten invariants of lens spaces and torus bundles, and the semiclassical approximation, Commun. Math. Phys.147 (1992) 563–604], we compute the quantum SU(N)-invariants for mapping tori of trace 2 homeomorphisms of a genus 1 surface when N = 2, 3 and discuss their asymptotics. In particular, we obtain directly a proof of a version of Witten's asymptotic expansion conjecture for these 3-manifolds. We further prove the growth rate conjecture for these 3-manifolds in the SU(2) case, where we also allow the 3-manifolds to contain certain knots. In this case we also discuss trace -2 homeomorphisms, obtaining — in combination with Jeffrey's results — a proof of the asymptotic expansion conjecture for all torus bundles.

An n-string tangle is a three-dimensional ball with n-strings properly embedded in it. In the late 1980s, Ernst and Sumners introduced a tangle model for protein-DNA complexes. The protein is modeled by a three-dimensional ball and the protein-bound DNA is modeled by strings embedded inside the ball. Originally the tangle model was applied to proteins such as Tn3 resolvase which binds two DNA segments. This protein breaks and rejoins two DNA segments and can create knotted DNA. A 2-string tangle model can be used for this complex. More recently, Pathania, Jayaram and Harshey determined that the topological structure of DNA within a Mu protein complex consists of three DNA segments containing five crossings. Since Mu binds DNA sequences at three sites, this Mu protein-DNA complex can be modeled by a 3-string tangle. Darcy, Leucke and Vazquez analyzed Pathania et al.'s experimental results by using 3-string tangle analysis. There are protein-DNA complexes that involve four or more DNA sites. When a protein binds circular DNA at four sites, a protein-DNA complex can be modeled by a 4-string tangle with four loops outside of the tangle. We determine a biologically relevant 4-string tangle model. We also develop mathematics for solving 4-string tangle equations to determine the topology of DNA within a protein complex.

Given an orientable surface with boundary and a free homotopy class of a closed curve on this surface, we present a purely combinatorial algorithm which produces a representative of that homotopy class with minimal self-intersection.