Journal of Knot Theory and Its Ramifications

An invariant polynomial s(t) is defined for framed knots in the solid torus. The coefficients are Vassiliev invariants of order one. An invariant polynomial A(t) of Legendrian curves is introduced and it is shown how to calculate it from their fronts. The coefficient of A(t) of the order n term is the restriction to the discriminant of the selftangencies with partial index n of the Arnold invariant J⁺ of wave fronts. The polynomial A(t) of a Legendrian curve is recovered from the polynomial s(t) of the Legendrian knot, provided with its natural contact framing.

Vassiliev. invariants up to order six for arbitrary torus knots {n, m}, with n and m coprime integers, are computed. These invariants are polynomials in n and m whose degree coincide with their order. Furthermore, they turn out to be integer-valued in a normalization previously proposed by the authors.

An explicit formulation of a third order finite knot invariant is derived from the perturbative expression of the Wilson loop integral along a knotted line in Chern-Simons theory. This is achieved by an appropriate deformation of the knot line in three dimensional space. It is demonstrated that this formulation fulfils the axioms of the Vassiliev invariants. We use our formula in order to calculate the invariant for knots with up to nine crossings and for some torus knots.

We construct a generalization of the “free group valued” Tabachnikov’s invariant of long curves. We characterize Tabachnikov’s polynomial invariant of plane curves via a representation of the twin group. We also show that these invariants are combinations of finite type invariants.

This paper proves that the set of all smooth, closed space curves having the same knot type and writhe is path-connected.

In [5] Xiao Song Lin shows that “traceless” representations of a braid group into SU(2) counted with appropriate sign gives one half of the signature of the associated knot. This is done with a skein technique, comparing the behavior of the representations and the sign of the Conway polynomial of the knot evaluated at 2i. These results appear somewhat mysterious.

In this paper, we present a representation of the braid group into SL(2n,Z) where n+1 is the number of strands of the braid. This matrix arises naturally from considering the two-fold branched cover of a ball containing the braid. The lower right n×n block is basically the reduced Brauer representation evaluated at –1 while the lower left n×n block of this matrix has special significance. The absolute value of the determinant of this block corresponds to the determinant of the corresponding closed braid. If one defines C(η) to be the determinant of this block normalized properly one gets an invariant of the corresponding closed braid which is equal to the Conway polynomial of the corresponding closed braid evaluated at 2i. Therefore, one can use a skein technique to compute the signature of the knot from C(η). Since the absolute value of the Conway polynomial evaluated at 2i is the order of the homology of the two-fold branched cover of a link, C(η) also contains this homological information. It will be shown in [6] that the lower left n×n block is a presentation matrix of the first homology of the two-fold cover. This homology matrix is related to the traceless representations of Xiao Song Lin that come from the homology of the two-fold branched cover.

For each odd number n, we describe a regular projection of a planar graph such that every spatial graph obtained by giving it over/under information of crossing points contains a (2, n)-torus knot. We also show that for any spatial graph H, there is a regular projection of a (possibly nonplanar) graph such that every spatial graph obtained from it contains a subgraph that is ambient isotopic to H.