“Pointed Quandle Coloring Quivers of Linkoids,” co-authored by Associate Professor of Mathematics Jose Ceniceros and Max Klivans ’25, was recently published in the Mediterranean Journal of Mathematics from Springer. In January, Klivans presented the work in Seattle at the Joint Mathematics Meetings (JMM), known as “the world’s largest annual mathematics conference.” It is organized by the American Mathematical Society.
Ceniceros summarized the paper, saying it “contributes to knot theory, a field of mathematics that studies how loops and entanglements behave in space.” He said that in order to study these objects, the authors enhanced “a previously defined linkoid invariant called the pointed quandle counting invariant.”
In the paper, Ceniceros and Klivans introduced quivers – diagrammatic structures that illustrate relationships between different ways of assigning elements of a pointed quandle to a given linkoid. Using these quivers, they defined a new invariant called the in-degree quiver polynomial matrix.
“This new polynomial invariant captures more structural information than the counting invariant alone, making it a strictly stronger tool for distinguishing linkoids,” Ceniceros said, adding that “these methods provide deeper insight into linkoid structure and reveal patterns that were not accessible using earlier techniques.” The paper also explores a family of linkoids related to torus links, applying these new tools to uncover meaningful mathematical relationships.
“This work highlights the creativity of mathematical exploration and the value of collaboration between faculty and undergraduate students,” Ceniceros said.
Klivans plans to start a Ph.D. program in mathematics this fall.
