Link invariants from finite racks

The Column Group and Its Link Invariants

“…where q (w1,...,wc) = c k=1 q w k . This enhancement keeps track of which writhe vectors contribute which colorings, and for certain racks determines the linking number mod N for links with two components [16].…”

“…Other examples of enhancements are known in special cases, such as quandle/biquandle/rack 2-cocycle enhancements [2,3,16], quandle/rack/biquandle polynomials [14,15,4], and various enhancements which use extra structure of the labeling objects, e.g. symplectic quandle enhancements [18] and Coxeter rack enhancements [19].…”

On Rack Polynomials

“…In [8], the quandle counting invariant |Hom(Q(L), T )| was extended to the case of finite non-quandle racks. In this section we will enhance this invariant with rack polynomials.…”

“…If two ambient isotopic diagrams of D have writhe vectors which are componentwise congruent modulo N (T ), then there is a bijection φ : Hom(F R(D, w), T ) → Hom(F R(D, w ), T ) between the sets of rack homomorphisms from the fundamental racks of (D, w) and (D, w ) into T defined by sending a coloring of one diagram to a coloring of the same diagram with mN kinks added. Indeed, since any subrack containing an element x ∈ T must also contain the rack powers x n for all n ∈ Z (see [8]), φ preserves image subracks. Hence, as far as T is concerned, the framing vectors of D live in W = (Z N (T ) ) c , and we have an invariant of unframed links given by We would like to jazz up these rack counting invariants with the generalized rack polynomials.…”

“…We are able to show that for at least one class of finite racks, the generalized rack polynomials determine the rack structure up to isomorphism, and we identify another class of quandles for which the generalized rack polynomials do not determine the isomorphism class. We then use these rack polynomials to enhance the rack counting invariants from [8].…”