Journal of Knot Theory and Its Ramifications

Recently, Stoimenow [J. Knot Th. Ram. 7 (1998), 93–114] gave an upper bound on the dimension d_n of the space of order n Vassiliev knot invariants, by considering chord diagrams of a certain type. We present a simpler argument which gives a better bound on the number of these chord diagrams, and hence on d_n.

We discuss the homotopy surface category of a space which generalizes the 1+1-dimensional cobordism category of circles and surfaces to the situation where one introduces a background space. We explain how for a simply connected background space, monoidal functors from this category to vector spaces can be interpreted in terms of Frobenius algebras with additional structure.

The Hopf link, consisting of two unknots wrapped around each other, is the simplest possible nontrivial link with more than one component. We can generalize it to two bundles of "parallel" unknots wrapped around each other. In this paper, we show that when one of the two bundles has a fixed side, the HOMFLY polynomials of the links satisfy a system of recurrence equations. This leads to a procedure for computing explicit formulas for the HOMFLY polynomials.

For knots in S³ criteria for free periodicity are obtained in terms of the bracket and the Jones polynomials. The criteria introduced in this paper generalize the result obtained by Murasugi for the Jones polynomials of periodic knots.

A differential geometric characterization of the braid-index of a link is found. After multiplication by 2π, it equals the infimum of the sum of total curvature and total absolute torsion over holonomic representatives of the link.

Upper and lower bounds for the infimum of the total curvature over holonomic representatives of a link are given in terms of its braid- and bridge-index. Examples showing that these bounds are sharp are constructed.

We calculate the highest and the lowest degrees of the Kauffman bracket polynomials of certain inadequate pretzel links and show that there is a knot K such that c(K) – r_deg V_K=k for any nonnegative integer k, where c(K) is the crossing number of K and r_deg V_K is the reduced degree of the Jones polynomial V_K of K.

Given a properly embedded graph Γ in a ball B and a punctured sphere Σ properly embedded in B-Γ we examine the conditions on Γ that are necessary to assure that Σ is boundary parallel.

Let φ:D²→D² be an orientation preserving homeomorphism of the disk into itself, and Φ= {φ_t}_0≤t≤1 an isotopy with φ₀=id_D² and φ₁=φ. Then for a finite union of periodic orbits P of φ, the set is a link in D²×S¹. We say that φ induces all link types (for Φ) if there exists a homeomorphism h of D²×S¹ into a standardly embedded solid torus in the 3-sphere S³ such that any link L in S³ can be realized by a finite union of periodic orbits P_L of φ so that L and are equivalent. We will show that the Smale horseshoe and its second power do not induce all link types, but its third power does induce all link types.

We find approximations by Vassiliev invariants for the coefficients of the Jones polynomial and all specializations of the HOMFLY and Kauffman polynomials. Consequently, we obtain approximations of some other link invariants arising from the homology of branched covers of links.

It is shown by Y. Ohyama, K. Taniyama and S. Yamada that for any natural number n and any knot K, there are infinitely many unknotting number one knots, whose Vassiliev invariants of order less than or equal to n coincide with those of K. We analyze it for delta unknotting number, and obtain the following. For any natural number n and any oriented knots K and M with a₂(K)≠a₂(M) there are infinitely many knots J_m such that the delta distance between J_m and M coincide with |a₂(K)-a₂(M)| and whose Vassiliev invariants of order less than or equal to n coincide with those of K. Here a₂(K) is the second coefficient of the Conway polynomial of K.