The Column Group and Its Link Invariants

Abstract

The column group is a subgroup of the symmetric group on the elements of a finite blackboard 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.

Quantum enhancements and biquandle brackets

“…It is standard practice for enhancements of counting invariants to be expressed in "polynomial form" by writing elements of the multiset as exponents of a formal variable u with positive integer multiplicities as coefficients. We note that while strictly speaking this only defines a genuine (Laurent) polynomial in case R = Z, this notation in common in the literature -it was introduced with quandle cocycle invariants in [4] and has been standard ever since, see [2,3,5,6,9,10,14,15] for instance. The invariant written in this format contains the same information as the multiset version and has certain advantages; for instance, evaluation of Φ β X at u = 1 (using the rule 1 r = 1 for all r ∈ R) yields the cardinality of the multiset version of the invariant , i.e.…”

“…In particular, the number of biquandle colorings of an oriented knot or link diagram K by a finite biquandle X defines a nonnegative integer-valued invariant known as the biquandle counting invariant, denoted Φ Z X (K). An enhancement of Φ Z X is a generally stronger invariant from which Φ Z X can be recovered; enhancements have been studied in [2,4,10,15] to name just a few.…”

Quasi-trivial quandles and biquandles, cocycle enhancements and link-homotopy of pretzel links

“…Unlike the case of quandles, the actions of y ∈ X on X given by α y , β y are not automorphisms of X in general, but they are still permutations of the elements of X and generate a subgroup CG of the symmetric group on X known as the column group in [7]. The orbits of the action of CG on X partition X into disjoint orbit sub-biquandles analogously to the quandle case.…”