Abstract
In an earlier paper the first author defined a non-commutative –polynomial for knots in 3–space, using the colored Jones function. The idea is that the colored Jones function of a knot satisfies a non-trivial linear –difference equation. Said differently, the colored Jones function of a knot is annihilated by a non-zero ideal of the Weyl algebra which is generalted (after localization) by the non-commutative –polynomial of a knot.
In that paper, it was conjectured that this polynomial (which has to do with representations of the quantum group ) specializes at to the better known –polynomial of a knot, which has to do with genuine representations of the knot complement.
Computing the non-commutative –polynomial of a knot is a difficult task which so far has been achieved for the two simplest knots. In the present paper, we introduce the –polynomial of a knot, along with its non-commutative version, and give an explicit computation for all twist knots. In a forthcoming paper, we will use this information to compute the non-commutative –polynomial of twist knots. Finally, we formulate a number of conjectures relating the , the –polynomial and the Alexander polynomial, all confirmed for the class of twist knots.
Citation
Stavros Garoufalidis. Xinyu Sun. "The $C$–polynomial of a knot." Algebr. Geom. Topol. 6 (4) 1623 - 1653, 2006. https://doi.org/10.2140/agt.2006.6.1623
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