Coloring n-String Tangles

This expository paper describes how the knot invariant Fox coloring [3,8] can be applied to tangles. An n-string tangle T is a 3-ball with n strings properly embedded in the 3-ball. The boundary of the 3-ball and the 2n endpoints of the n-strings on the boundary of the 3-ball are not allowed to move. Tangles were rst used by John H. Conway to tabulate knots [10]. Following the presentation in [7], we will describe coloring via systems of linear equations so that only an introductory background in linear algebra will be needed. Fox coloring is related to many beautiful areas in topology. Our interest in this method of coloring links and tangles is to make this paper accessible to nonmathematicians as this method is used computationally to solve tangle equations arising from protein-DNA interactions [2]. Also, this approach can make open problems in this area accessible to undergraduates. For example, the results of [11,6] can be proved using only this linear algebra denition of Fox coloring combined with a neat trick of Przytycki [8]. We will begin with a brief review on coloring knots/links in section 2. In this section we will provide examples, but no proofs. For proofs see [7]. Most of the proofs for knots/links are also similar to those for tangles given in section 3. In section 4, we extend the coloring denition to tangles containing a nite number of circles. In section 5, we give some formulas for determining these invariants for 3-string braids and 2-string rational tangles. In sections 7 and 8, we discuss embedding tangles in knots. We make some concluding remarks in section 9.