Narrow Results

We introduce an algebra ℤ[X, S] associated to a pair X, S of a virtual birack X and X-shadow S. We use modules over ℤ[X, S] to define enhancements of the virtual birack shadow counting invariant, extending the birack shadow module invariants to virtual case. We repeat this construction for the twisted virtual case. As applications, we show that the new invariants can detect orientation reversal and are not determined by the knot group, the Arrow polynomial and the Miyazawa polynomial, and that the twisted version is not determined by the twisted Jones polynomial.

We introduce augmented biracks and define a (co)homology theory associated to augmented biracks. The new homology theory extends the previously studied Yang–Baxter homology with a combinatorial formulation for the boundary map and specializes to N-reduced rack homology when the birack is a rack. We introduce augmented birack 2-cocycle invariants of classical and virtual knots and links and provide examples.

The column group is a subgroup of the symmetric group on the elements of a finite birack generated by the column permutations in the birack matrix. We use subgroups of the column group associated to birack homomorphisms to define an enhancement of the integral birack counting invariant and give examples which show that the enhanced invariant is stronger than the unenhanced invariant.

We introduce an associative algebra ℤ[X, S] associated to a birack shadow and define enhancements of the birack counting invariant for classical knots and links via representations of ℤ[X, S] known as shadow modules. We provide examples which demonstrate that the shadow module enhanced invariants are not determined by the Alexander polynomial or the unenhanced birack counting invariants.

The involutory birack counting invariant is an integer-valued invariant of unoriented tangles defined by counting homomorphisms from the fundamental involutory birack of the tangle to a finite involutory birack over a set of framings modulo the birack rank of the labeling birack. In this first of an anticipated series of several papers, we enhance the involutory birack counting invariant with quantum weights, which may be understood as tangle functors of involutory birack-labeled unoriented tangles.

We consider birack and switch colorings of braids. We define a switch structure on the set of permutation representations of the braid group and consider when such a representation is a switch automorphism. We define quiver-valued invariants of braids using finite switches and biracks and use these to categorify the birack 2-cocycle invariant for braids. We obtain new polynomial invariants of braids via decategorification of these quivers.

We extend the rack algebra ℤ[X] defined by Andruskiewitsch and Graña to the case of biracks, enabling a notion of birack modules. We use these birack modules to define an enhancement of the birack counting invariant generalizing the birack module counting invariant in [A. Haas, G. Heckel, S. Nelson, J. Yuen and Q. Zhang, Rack module enhancements of counting invariants, Osaka J. Math.49 (2012) 471–488]. We provide examples demonstrating that the enhanced invariant is not determined by the Jones or Alexander polynomials and is stronger than the unenhanced birack counting invariant.

Biracks are algebraic structures related to knots and links. We define a new enhancement of the birack counting invariant for oriented knots and links via algebraic structures called birack dynamical cocycles. The new invariants can also be understood in terms of partitions of the set of birack labelings of a link diagram determined by a homomorphism p : X → Y between finite labeling biracks. We provide examples to show that the new invariant is stronger than the unenhanced birack counting invariant and examine connections with other knot and link invariants.