Chien, Julien* and Sam Nelson. “Virtual Links with Finite Medial Bikei.” Journal of Symbolic Computation, vol. 92, 2019, pp. 211-221.
Abstract: We consider the question of which virtual knots have finite fundamental medial bikei. We describe and implement an algorithm for completing a presentation matrix of a medial bikei to an operation table, determining both the cardinality and isomorphism class of the fundamental medial bikei, each of which are link invariants. As an example, we compute the fundamental medial bikei for all of the prime virtual knots with up to four classical crossings as listed in the knot atlas.
Cho, Karina*, and Sam Nelson. “Quandle Cocycle Quivers.” Topology and Its Applications, vol. 268, 2019, 106908.
Abstract: We incorporate quandle cocycle information into the quandle coloring quivers we defined in  to define weighted directed graph-valued invariants of oriented links we call quandle cocycle quivers. This construction turns the quandle cocycle invariant into a small category, yielding a categorification of the quandle cocycle invariant. From these graphs we define several new link invariants including a 2-variable polynomial which specializes to the usual quandle cocycle invariant. Examples and computations are provided.
Cho, Karina*, and Sam Nelson. “Quandle Coloring Quivers.” Journal of Knot Theory and Its Ramifications, vol. 29, no. 1, 2019, 1950001.
Abstract: We consider a quiver structure on the set of quandle colorings of an oriented knot or link diagram. This structure contains a wealth of knot and link invariants and provides a categorification of the quandle counting invariant in the most literal sense, i.e. giving the set of quandle colorings the structure of a small category which is unchanged by Reidemeister moves. We derive some new enhancements of the counting invariant from this quiver structure and show that the enhancements are proper with explicit examples.
Gügümcü, Neslihan, and Sam Nelson. “Biquandle Coloring Invariants of Knotoids.” Journal of Knot Theory and Its Ramifications, vol. 28, no. 4, 2019, 1950029.
Abstract: In this paper, we consider biquandle colorings for knotoids in ℝ2 or S2, and we construct several coloring invariants for knotoids derived as enhancements of the biquandle counting invariant. We first enhance the biquandle counting invariant by using a matrix constructed by utilizing the orientation a knotoid diagram is endowed with. We generalize Niebrzydowski’s biquandle longitude invariant for virtual long knots to obtain new invariants for knotoids. We show that biquandle invariants can detect mirror images of knotoids and show that our enhancements are proper in the sense that knotoids which are not distinguished by the counting invariant are distinguished by our enhancements.
Nelson, Sam. “A Survey of Quantum Enhancements.” Knots, Low-Dimensional Topology and Applications: Knots in Hellas, International Olympic Academy, Greece, July 2016, edited by Colin C. Aams, Cameron McA. Gordon, Vaughan F.R. Jones, Louis H. Kauffman, Sofia Lambropoulou, Kenneth C. Millett, Jozaef H. Przytycki,, Renzo Ricca, and Radmila Sazdanovic. Springer, 2019, pp. 163-178.
Abstract: In this short survey article, we collect the current state of the art in the nascent field of quantum enhancements, a type of knot invariant defined by collecting values of quantum invariants of knots with colorings by various algebraic objects over the set of such colorings. This class of invariants includes classical skein invariants and quandle and biquandle cocycle invariants as well as new invariants.
Nelson, Sam, Kanako Oshiro, and Natsumi Oyamaguchi. “Local Biquandles and Niebrzydowski’s Tribracket Theory.” Topology and Its Applications, vol. 258, 2019, pp. 474-512.
Abstract: We introduce a new algebraic structure called local biquandles and show how colorings of oriented classical link diagrams and broken surface diagrams are related to tribracket colorings. We define a (co)homology theory for local biquandles and show that it is isomorphic to Niebrzydowski’s tribracket (co)homology. This implies that Niebrzydowski’s (co)homology theory can be interpreted similarly as biquandle (co)homology theory. Moreover through the isomorphism between two cohomology groups, we show that Niebrzydowski’s cocycle invariants and local biquandle cocycle invariants are the same.
Nelson, Sam, Kanako Oshiro, Ayaka Shimizu, and Yoshiro Yaguchi. “Biquandle Virtual Brackets.” Journal of Knot Theory and Its Ramifications, vol. 28, no. 11, 2019, 1940003.
Abstract: We introduce an infinite family of quantum enhancements of the biquandle counting invariant which we call biquandle virtual brackets. Defined in terms of skein invariants of biquandle colored oriented knot and link diagrams with values in a commutative ring R using virtual crossings as smoothings, these invariants take the form of multisets of elements of R and can be written in a “polynomial” form for convenience. The family of invariants defined herein includes as special cases all quandle and biquandle 2-cocycle invariants, all classical skein invariants (Alexander–Conway, Jones, HOMFLYPT and Kauffman polynomials) and all biquandle bracket invariants defined in [S. Nelson, M. E. Orrison and V. Rivera, Quantum enhancements and biquandle brackets, J. Knot Theory Ramifications 26(5) (2017) 1750034] as well as new invariants defined using virtual crossings in a fundamental way, without an obvious purely classical definition.
Nelson, Sam, and Shane Pico*. “Virtual Tribrackets.” Journal of Knot Theory and Its Ramifications, vol. 28, no. 4, 2019, 1950026.
Abstract: We introduce virtual tribrackets, an algebraic structure for coloring regions in the planar complement of an oriented virtual knot or link diagram. We use these structures to define counting invariants of virtual knots and links and provide examples of the computation of the invariant; in particular, we show that the invariant can distinguish certain virtual knots.