Year: 2025
Type: article
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 [Formula: see text] along its axis of symmetry. If both J and [Formula: see text] are partial knots for different symmetric union presentations of the same ribbon knot K, the knots J and [Formula: see text] are said to be symmetrically related. Lamm proved that if J 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 m 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 J is the partial knot in a symmetric union presentation for K, then for any odd prime p we have [Formula: see text], where [Formula: see text] denotes the number of p-colorings of a knot.
Authors: Ben Clingenpeel, Zongzheng Dai, Gabriel Diraviam, Kareem Jaber, Krishnendu Kar, Ziyun Liu, Teo Miklethun, Haritha Nagampoozhy, Michael Perry, Moses Samuelson-Lynn, Eli Seamans, Ana Wright, Nicole Xie, Ruiqi Zou, Alexander Zupan
Institutions: George Washington University, Rensselaer Polytechnic Institute, University of Virginia, Princeton University, Louisiana State University, Carnegie Mellon University, University of Michigan, Indian Institute of Science Education and Research Pune, Syracuse University, Max Planck Institute for Mathematics in the Sciences, Davidson College, University of Nebraska–Lincoln
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G6160492840, G8274484846, G8315732210
OpenAlex ID: W4415737798