Quandle cohomology and state-sum invariants of knotted curves and surfaces

Abstract

Abstract. The 2-twist spun trefoil is an example of a sphere that is knotted in 4-dimensional space. A proof is given in this paper that this sphere is distinct from the same sphere with its orientation reversed. Our proof is based on a state-sum invariant for knotted surfaces developed via a cohomology theory of racks and quandles (also known as distributive groupoids).A quandle is a set with a binary operation -the axioms of which model the Reidemeister moves in classical knot theory. Colorings of diagrams of knotted curves and surfaces by quandle elements, together with cocycles of quandles, are used to define state-sum invariants for knotted circles in 3-space and knotted surfaces in 4-space.Cohomology groups of various quandles are computed herein and applied to the study of the state-sum invariants. Non-triviality of the invariants is proved for a variety of knots and links, and conversely, knot invariants are used to prove non-triviality of cohomology for a variety of quandles.

Knot colouring polynomials

“…Lemma 3.20 [Carter et al 1999;2003b]. The state-sum S λ Q is invariant under Reidemeister moves and thus defines a knot invariant S λ Q : → ‫ޚ‬ .…”

“…Every quandle colouring number F q Q is the specialization of some knot colouring polynomial P x G . Quandle cohomology was initially studied in order to construct invariants in low-dimensional topology: in [Carter et al 1999;2003b] it was shown how a 2-cocycle λ ∈ Z 2 (Q, ) gives rise to a state-sum invariant of knots, S λ Q : → ‫ޚ‬ , which refines the quandle colouring number F Q . We prove the following result: Theorem 1.11 (Section 3E).…”

“…Quandle cohomology was studied in [Carter et al 1999;2003b], where it was shown how a 2-cocycle gives rise to a state-sum invariant of knots in ‫ޓ‬ 3 . We recall this construction in Section 3D and show that every colouring polynomial P x G can be presented as a quandle 2-cocycle state-sum invariant, provided that the subgroup = C(x) ∩ G is abelian (Theorem 3.24).…”

Generalized Quandle Polynomials

“…For example, Joyce [7] shows that the square knot and the granny knot have nonisomorphic knot quandles despite having isomorphic knot groups. 2 In [11] we have the following definition.…”

Sequences of Knots and Their Limits

“…One way of implementing these ideas is via the CJKLS invariant of knots ( [1,2]). There is a CJKLS invariant for each choice of labeling quandle X , abelian group A, and 2-co-cycle φ in H 2 (X , A).…”