Colorings of symmetric unions and partial knots

Motivated by work of Kinoshita and Teraska, Lamm introduced the notion of a symmetric union, which can be constructed from a partial knot [Formula: see text] by introducing additional crossings to a diagram of [Formula: see text] along its axis of symmetry. If both [Formula: see text] and [Formula: see text] are partial knots for different symmetric union presentations of the same ribbon knot [Formula: see text], the knots [Formula: see text] and [Formula: see text] are said to be symmetrically related. Lamm proved that if [Formula: see text] and [Formula: see text] are symmetrically related, then [Formula: see text], asking whether the converse is true. In this paper, we give a negative answer to Lamm’s question, constructing for any natural number [Formula: see text] a family of [Formula: see text] 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 [Formula: see text] is the partial knot in a symmetric union presentation for [Formula: see text], then for any odd prime [Formula: see text] we have [Formula: see text], where [Formula: see text] denotes the number of [Formula: see text]-colorings of a knot.