On Conway’s potential function for colored links. (English) Zbl 1337.57025
Summary: The Conway potential function (CPF) for colored links is a convenient version of the multivariable Alexander-Conway polynomial. We give a skein characterization of CPF, much simpler than the one by Murakami. In particular, Conway’s “smoothing of crossings” is not in the axioms. The proof uses a reduction scheme in a twisted group-algebra \(\mathbb P_nB_n\), where \(B_n\) is a braid group and \(P_n\) is a domain of multi-variable rational fractions. The proof does not use computer algebra tools. An interesting by-product is a characterization of the Alexander-Conway polynomial of knots.
MSC:
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |
| 20F36 | Braid groups; Artin groups |
References:
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| [2] | Cimasoni, D.: A geometric construction of the Conway potential function. Comment. Math. Helv., 79(1), 124-146 (2004) · Zbl 1044.57002 · doi:10.1007/s00014-003-0777-6 |
| [3] | Conway, J. H., An enumeration of knots and links, and some of their algebraic properties, 329-358 (1970), New York · Zbl 0202.54703 |
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| [5] | Murakami, J., On local relations to determine the multi-variable Alexander polynomial of colored links, 455-464 (1992), New York · Zbl 0764.57008 |
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