Journal of Knot Theory and Its Ramifications

We show there is an infinite number of knots whose unknotting numbers can only be realized through a sequence of crossing changes on a nonminimal projection, if ambient isotopies between crossing changes are not allowed. In this process, the question arises one crossing change at a time is made, always on a minimal projection, in order to obtain the unknotting number of a knot.

We show that under twisting, a Vassiliev invariant of order n behaves like a polynomial of degree at most n. This greatly restricts the values that a Vassiliev invariant can take, for example, on the (2, m) torus knots. In particular, this implies that many classical numerical knot invariants such as the signature, genus, bridge number, crossing number, and unknotting number are not Vassiliev invariants.

The purpose of this paper is to consider the –representation space of knot groups in order to define a polynomial invariant of the knot. Moreover, we would like to study these invariants in special cases and to prove that the polynomial is not trivial if the knot has a non-trivial longitude invariant.

It is proved that any two mutative closed oriented 3-manifolds have the same quantum SU(2) invariant. By a constructive argument of topological imitation, we construct finitely many mutative hyperbolic imitations of any given closed oriented 3-manifold with certain arbitrariness of isometry groups whose quantum SU(2) invariants are close to the original one.

We show that every non-trivial tame knot or link in ℝ³ has a quadrisecant, i.e. four collinear points. The quadrisecant must be topologically non-trivial in a precise sense. As an application, we show that a nonsingular, algebraic surface in ℝ³ which is a knotted torus must have degree at least eight.

We give the definition of the Partition Algebra P_n(Q). This is a new generalisation of the Temperley–Lieb algebra for Q-state n-site Potts models, underpinning their transfer matrix formulation on arbitrary transverse lattices. In P_n(Q) subalgebras appropriate for building the transfer matrices for all transverse lattice shapes (e.g. cubic) occur. For the Partition algebra manifests either a semi-simple generic structure or is one of a discrete set of exceptional cases. We determine the Q-generic and Q-independent structure and representation theory. In all cases (except Q = 0) simple modules are indexed by the integers j ≤ n and by the partitions λ ˫ j. Physically they may be associated, at least for sufficiently small j, to 2j 'spin' correlation functions.

We exhibit a subalgebra isomorphic to the Brauer algebra.

Templates are used to capture the knotting and linking patterns of periodic orbits of positive entropy flows in 3 dimensions. Here, we study the properties of various templates, especially whether or not there is a bound on the number of prime factors of the knot types of the periodic orbits. We will also see that determining whether two templates are different is highly nontrivial.