Colorings of symmetric unions and partial knots

Abstract

Motivated by work of Kinoshita and Teraska, Lamm introduced the notion of a symmetric union, which can be constructed from a partial knot J by introducing additional crossings to a diagram of J# −J along its axis of symmetry. If both J and J′ are partial knots for different symmetric union presentations of the same ribbon knot K, the knots J and J′ are said to be symmetrically related. Lamm proved that if J and J′ are symmetrically related, then det J = det J′, asking whether the converse is true. In this paper, we give a negative answer to Lamm’s question, constructing for any natural number m a family of 2m knots with the same determinant but such that no two knots in the family are symmetrically related. This result is a corollary to our main theorem, that if J is the partial knot in a symmetric union presentation for K, then for any odd prime p we have (Formula presented), where colp(·) denotes the number of p-colorings of a knot.