Pure and Applied Mathematics Quarterly

Volume 11 (2015)

Number 3

Calculating colored homflypt invariants with image processing and a new form of Gauss code

Pages: 503 – 535

DOI: http://dx.doi.org/10.4310/PAMQ.2015.v11.n3.a5

Author

Xin Zhou (Center of Mathematical Sciences, Zhejiang University, Hangzhou, China)

Abstract

We propose a new form of signed Gauss Code consisting of a list of matrices, named as intrinsic matrices (IMAT). We use IMAT to calculate colored homflypt invariants in three steps. Firstly, given a link diagram, we obtain its IMAT by image processing. Secondly, if the link is decorated by the idempotent basis, we acquire an IMAT expression for the decorated link by using the IMAT obtained in the first step. This expression is a linear combination of a few IMATs. Finally, we improve the skein template algorithm and apply it to each IMAT in the expression got in the second step, which leads to the colored homflypt invariant of the given idempotent decorated link. The procedure is illustrated with several examples. The first two steps are the main new ingredients in the scheme and it is by using them that we are able to acquire the IMAT expression for the decorated link and compute the colored homflypt invariants for arbitrary link with more intuitive inputs.

Keywords

homflypt invariant, Gauss Code, intrinsic matrices (IMAT); image processing, skein template algorithm, decorated link

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