Churchill, Indu R.U., Mohamed Elhamdadi, Mustafa Hajij, and Sam Nelson. "Singular knots and involutive quandles." Journal of Knot Theory and Its Ramifications, vol. 26, no. 14, 2017, 1750099.
Abstract: The aim of this paper is to define certain algebraic structures coming from generalized Reidemeister moves of singular knot theory. We give examples, show that the set of colorings by these algebraic structures is an invariant of singular links. As an application we distinguish several singular knots and links.
Ishii, Atsushi, and Sam Nelson. "Partially Multiplicative Biquandles and Handlebody-Knots." Contemporary Mathematics, vol. 689, 2017, pp. 159-176.
Abstract: We introduce several algebraic structures related to handlebody-knots, including G-families of biquandles, partially multiplicative biquandles and group decomposable biquandles. These structures can be used to color the semiarcs in Y-oriented spatial trivalent graph diagrams representing S1-oriented handlebody-knots to obtain computable invariants for handlebody-knots and handlebody-links. In the case of G-families of biquandles, we enhance the counting invariant using the group G to obtain a polynomial invariant of handlebody-knots.
Needell, Deanna, and Sam Nelson. "Biquasiles and Dual Graph Diagrams." Journal of Knot Theory and Its Ramifications, vol. 26, no. 8, 2017, 1750048.
Abstract: We introduce dual graph diagrams representing oriented knots and links. We use these combinatorial structures to define corresponding algebraic structures we call biquasiles whose axioms are motivated by dual graph Reidemeister moves, generalizing the Dehn presentation of the knot group analogously to the way quandles and biquandles generalize the Wirtinger presentation. We use these structures to define invariants of oriented knots and links and provide examples.
Nelson, Sam, Michael E. Orrison, and Veronica Rivera. "Quantum Enhancements and Biquandle Brackets." Journal of Knot Theory and Its Ramifications, vol. 26, no. 05, 2017, 1750034.
Abstract: We introduce a new class of quantum enhancements we call biquandle brackets, which are customized skein invariants for biquandle colored links.Quantum enhancements of biquandle counting invariants form a class of knot and link invariants that includes biquandle cocycle invariants and skein invariants such as the HOMFLY-PT polynomial as special cases, providing an explicit unification of these apparently unrelated types of invariants. We provide examples demonstrating that the new invariants are not determined by the biquandle counting invariant, the knot quandle, the knot group or the traditional skein invariants.
Nelson, Sam, and Jake Rosenfield. "Bikei Homology." Homology, Homotopy, and Applications, vol. 19, no. 1, 2017, pp. 23-35.
Abstract: We introduce a modified homology and cohomology theory for involutory biquandles (also known as bikei). We use bikei 2-cocycles to enhance the bikei counting invariant for unoriented knots and links as well as unoriented and non-orientable knotted surfaces in ℝ4.
Nelson, Sam, and Natsumi Oyamaguchi. “Trace Diagrams and Biquandle Brackets.” International Journal of Mathematics, vol. 28, no. 14, 2017, 1750104.
Abstract: We introduce a method of computing biquandle brackets of oriented knots and links using a type of decorated trivalent spatial graphs we call trace diagrams. We identify algebraic conditions on the biquandle bracket coeﬃcients for moving strands over and under traces and identify a new stop condition for the recursive expansion. In the case of monochromatic crossings we show that biquandle brackets satisfy a Homﬂypt-style skein relation and we identify algebraic conditions on the biquandle bracket coeﬃcients to allow pass-through trace moves.