Base-Tangle Decompositions of $n$-String Tangles with $1<n<10$

Abstract

This study describes the program bTd, which was developed for the decomposition of any $n$-tangle with $1 < n < 10$} into base {\small$n$}-tangles using the Skein relation. The program enables us to compute HOMFLY polynomials of knots and links with a large number of crossing points within a matter of hours (see Examples 4.4 and 4.5). This contrasts with the results of attempting computations using Hecke algebras $H(q,n)$} with $18 \ge n$. Such a computation did not complete even after a period of thirty days in a recent examination by the first author and F. Kako [Imafuji and Ochiai 02, Murakami 89, Ochiai and Murakami 94, Ochiai and Kako 95]. In this paper, we first introduce two new concepts: an oriented ordered tangle and a subdivision of a tangle. We then present some examples of base-tangle decompositions achieved using the present program along with the corresponding computational times.