Kauffman-Harary conjecture holds for Montesinos Knots

Details Kauffman-Harary conjecture holds for Montesinos Knots Kauffman-Harary conjecture holds for Montesinos Knots

2003-05-29

arxiv.org/abs/math/0305415v1

Mathematics

Geometric Topology

to appear in Journal of Knot Theory and Ramifications, 11 pages, 10 figures

Scientific paper

The Kauffman-Harary conjecture states that for any reduced alternating diagram K of a knot with a prime determinant p, every non-trivial Fox p-coloring of K assigns different colors to its arcs. We generalize the conjecture by stating it in terms of homology of the double cover of S^3 branched along a link. In this way we extend the scope of the conjecture to all prime alternating links of arbitrary determinants. We first prove the Kauffman-Harary conjecture for pretzel knots and then we generalize our argument to show the generalized Kauffman-Harary conjecture for all Montesinos links. Finally, we speculate on the relation between the conjecture and Menasco's work on incompressible surfaces in exteriors of alternating links.

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