## Base-tangle decompositions of $$n$$-string tangles with $$1< n< 10$$.(English)Zbl 1153.57009

Summary: This study describes the program bTd, which was developed for the decomposition of any $$n$$-tangle with $$1 < n < 10$$ into base $$n$$-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 $$18 \geq n$$. Such a computation did not complete even after a period of thirty days in a recent examination by the first author and F. Kako [Exp. Math. 4, No. 1, 62–67 (1995; Zbl 0846.20016)]. 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.

### MSC:

 57M25 Knots and links in the $$3$$-sphere (MSC2010) 57N10 Topology of general $$3$$-manifolds (MSC2010) 57-04 Software, source code, etc. for problems pertaining to manifolds and cell complexes

### Keywords:

knot; tangle decomposition; polynomial invariant; mutation

Zbl 0846.20016

K2K
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