Journal of Knot Theory and Its Ramifications

By using the Seiberg-Witten invariant of 3-dimensional manifolds, we obtain an estimate of genus of links in the three shere.

We show how to build tangles T in a 3-ball with the property that any knot obtained by tangle sum with T has a persistent lamination in its exterior, and therefore has property P. The construction is based on an example of a persistent lamination in the exterior of the twist knot 6₁, due to Ulrich Oertel. We also show how the construction can be generalized to n-string tangles.

We show that there is a family of hyperbolic knots each member of which admits a toroidal surgery and for which the collection of tunnel numbers (hence volumes also) is unbounded. As these are knots in the 3-sphere, the collection of cusp volumes of this family is bounded.

We study the first common terms of the HOMFLY and Kauffman polynomials of a knot, which we call the ϒ-polynomial, and the Conway polynomial for the 2-bridge knots and a class of 3-bridge knots. We characterize the ϒ-polynomial using the 2-bridge knots. Then we give some relations between the two polynomial invariants, and as an application, we consider the space of the Vassiliev invariant of order four.

A result of Milnor [1] states that the infimum of the total curvature of a tame knot K is given by 2πμ(K), where μ(K) is the crookedness of the knot K. It is also known that μ(K)=b(K), where b(K) is the bridge index of K [2]. The situation appears to be quite different for knots realised as polygons in the cubic lattice. We study the total curvature of lattice knots by developing algebraic techniques to estimate minimal curvature in the cubic lattice. We perform simulations to estimte the minimal curvature of lattice knots, and conclude that the situation is very different than for tame knots in ℛ³.

Simple formulas involving only linking numbers and surgery coefficients are presented for computing how the Casson-Walker invariant changes under crossing changes in a framed link presenting a 3-manifold. These formulas greatly simplify computations of the Casson-Walker invariant and can be used to define a type two polynomial invariant of unframed links.

We compute Kauffman's potential function (equivalent to the Conway polynomial) for any solvable fibered link. Solvable links are links that can be built from the unknot by iterated cabling and summing. Links that arise in complex algebraic geometry are generally of this type.

We use the new approach of braiding sequences and the Stanford construction to prove that there is no way to extract information on fiberedness from Vassiliev invariants and that the chirality sensitivity of Vassiliev invariants depends mainly only on the parity of their degree.