Jose Ceniceros

Anthony ChristianaAssociate Professor of Mathematics Jose Ceniceros and Anthony Christiana ’22 co-authored a paper that was recently published in the Journal of Knot Theory and Its Ramifications.

Written with Sam Nelson of Claremont McKenna College, the paper presents what the authors call “a new and improved method for classifying singular knots and pseudoknots.” Basing their approach on a mathematical structure known as the psyquandle, they introduced psyquandle coloring quivers to enhance the psyquandle counting invariant. Ceniceros said the methodology was “inspired by the quandle coloring quivers used to classify classical knots, which employ an object from representation theory.”

He noted that using the psyquandle coloring quiver to extend the in-degree polynomial invariants derived from the theoretical framework of quandle coloring quivers is a significant development, as it broadens the applicability of these invariants to include the complex structures of singular knots and pseudoknots.

“An unexpected outcome of defining the psyquandle coloring quiver was the emergence of biquandle coloring quivers,” he added. “These quivers are a direct generalization of quandle coloring quivers and provide us with a better understanding of both classical and virtual knots and links. This article introduces several tools to study a broad range of knot objects.”

In 2022, Christiana completed a senior honors project on this topic. He is currently pursuing a Ph.D. in mathematics at The George Washington University.

Ceniceros plans to offer his senior seminar on knot theory again next fall.

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