2-adjacency between knots. (English) Zbl 1327.57013
Summary: We study the properties of a knot \(K\) to be 2-adjacent to another knot W by analyzing their Conway polynomials, Jones polynomials and Homfly polynomials and give some very useful conditions. We discuss whether each pair of knots can be 2-adjacent to each other, i.e. whether 2-adjacency is a symmetric relation. We discuss also whether the trivial knot, the trefoil knot and the figure-eight knot can be 2-adjacent to any knot in Rolfsen’s table and the opposite cases, except for \(9_{34}\) it is not decided whether it is 2-adjacent to \(4_{1}\). Finally, we give some examples to answer I. Torisu’s problem partly and etc.
MSC:
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |
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