An article co-authored by Assistant Professor of Mathematics Jose Ceniceros was recently published online by the Canadian Mathematical Bulletin. “Singquandle Shadows and Singular Knot Invariants” was written with Indu R. Churchill of SUNY Oswego and Mohamed Elhamdadi of the University of South Florida.
The article is the third paper in a series in which the authors use the singquandle algebraic structure to study singular knot. Ceniceros said this research aims to define effective and computable invariants of singular knots and links.
They first defined the notion of a singquandle action on a set to define the shadow counting invariant of singular knots and links. Using the singquandle action on a set, they then defined a polynomial invariant of singquandle shadow structures, which they call the singquandle shadow polynomial.
Next, they showed that the shadow polynomial invariant is an effective invariant for classifying singquandle shadow structures. Finally, they enhanced the shadow counting invariant of singular knots and links by combining the shadow counting invariant and shadow polynomial invariant to obtain the singquandle shadow polynomial invariant of singular knots and links.
The authors were then able to show “that the singquandle shadow polynomial invariant of singular knots and links detects knot information not detected by the singquandle counting invariant, the shadow counting invariant, and the singquandle polynomial invariant.
“Therefore,” they said, “the singquandle shadow polynomial is a more robust tool for classifying singular knots and links than the previously defined invariants derived from the singquandle algebraic structure.”