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.


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


Zbl 0846.20016


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