N. Nemirovskaya, Department of Mathematics
First of all I would like to thank the Office of Research and Creative Activities for choosing me as one of the recipients of the Research and Creative Activities scholarship. This scholarship allowed me to spend more time working on the proposed project.
This project is devoted to the computation of polynomial invariants of torus links in three-dimensional space. Roughly speaking, a link is a union of non-intersecting closed smooth curves in the space. A knot is a link with one connected component (i.e. a closed smooth curve in the space). A torus link (resp. torus knot) is a link (resp. knot) which can be drawn on a torus.
An invariant of links is a function on the set of links which doesn’t change its value under smooth deformations of links. A polynomial invariant of links is an invariant with values in the ring of polynomials of one or several independent variables. One of the most prominent advances in the mathematics of 80’s was the invention of new link invariants. The study of these invariants has lead to the discovery of deep connections between knot theory, mathematical physics and representation theory.
Torus links present a very interesting “experimental” material for the study of polynomial link invariants. In their paper “On the invariants of torus knots derived from quantum groups”, V. Jones and M. Rosso gave a general formula for the values of polynomial invariants associated with representations of quantum groups, on torus links. However, this formula involves some implicit terms which have to be computed for each particular case separately. In the case of a torus knot they show how to make this formula explicit for the classical series of quantum group and their representations corresponding to the so called Homfiy polynomial. I studied another particular case concerning the values of the Jones polynomial on an arbitrary torus link and have obtained the following explicit formula:
where Vm,n(t) is the Jones polynomial of the torus link with parameters n and m (which are positive integers), d is the greatest common divisor of m and n, and the summation is taken over all k such that 0 ~ k ~ n and km is divisible by n.
The method used by V. Jones and M. Rosso for the proof of their formula is quite technical which is adequate to the degree of its generality. However, in the case of the Jones polynomial there is a more direct combinatorial definition of the invariant due to L. Kauffman. The idea of such a definition came from statistical physics: each knot diagram has a number of “states” which correspond to resolving all intersections in two possible ways. With every state one associates a certain polynomial- its “weight.” Now the Jones polynomial of the original knot is computed as the sum of all the polynomials associated with all possible states. This approach suggests the problem of evaluating the invariant on concrete links by elementary means. In my work I applied Kauffman’s procedure of computing the Jones polynomial to torus links, and I have obtained an algebraic recursive algorithm for the calculation of the corresponding polynomials. The combinatorics of this algorithm turned out to be very involved. In several particular cases when the parameters n and m of a toric links satisfy some congruence relations, I derived explicit recursive formulas for the Jones polynomials. As the following example shows even in these relatively simple cases the result is quite nonwtrivial:
where it is assumed as part of definition that Vo,o = -(tl/2 +t-1/2 )-I and Vm,n = 0 if m < 0. I expect that the comparison of this and other similar formulas I obtained with (I) may lead to some interesting combinatorial identities.
I propose to continue this research and to expand it into my future Master thesis under the supervision of Dr. S. Humphries.