Computations of Quandle Cocycle Invariants of Knotted Curves and Surfaces

Abstract

State-sum invariants for knotted curves and surfaces using quandle cohomology were introduced by Laurel Langford and the authors (Quandle cohomology and state-sum invariants of knotted curves and surfaces, preprint). In this paper we present methods to compute the invariants and sample computations. Computer calculations of cohomological dimensions for some quandles are presented. For classical knots, Burau representations together with Maple programs are used to evaluate the invariants for knot table. For knotted surfaces in 4-space, movie methods and surface braid theory are used. Relations between the invariants and symmetries of knots are discussed. Academic Press

Quandle homotopy invariants of knotted surfaces

“…We now reformulate the (generalized) quandle cocycle invariants in [3,4,6] from our quandle homotopy invariant. We first remark that, if B X Y is path-connected, the inclusion (1) has no 1-cell by definition.…”

“…For an X -set Y , regarding the free module M = Z Y as a Z[As(X )]-module, the complexes (C R * (X ; M), ∂ * ) and (C Q * (X ; M), ∂ * ) are chain isomorphic to the cellular complexes of B X Y and of B X Y Q , respectively. As the simplest case, if Y is a single point, then (C R * (X ; Z), ∂ * ) and (C Q * (X ; Z), ∂ * ) coincide with the rack complex and the quandle complex of X described in [4], respectively.…”

“…Given a quandle X , Fenn et al [13] defined the rack space B X, in analogy to the classifying spaces of groups; Further, an invariant of framed links in S 3 was proposed in [14, §4] which is called a quandle homotopy invariant and is valued in the second homotopy group π 2 (B X) [see [23] for some computations of π 2 (B X)]. In addition, as a modification of the homology H * (B X; Z), Carter et al [4] introduced its quandle homology denoted by H Q n (X ; A), and further quandle cocycle invariants of classical links (resp. linked surfaces) using cohomology classes in H 2 Q (X ; A) (resp.…”

Homotopical interpretation of link invariants from finite quandles

“…As is known [RS,N1,CKS], given a quandle 2-cocycle φ : X 2 → A, the pairing between this φ and the cycle invariant Φ X (L) coincides with the original cocycle invariant in [CJKLS,Theorem 4.4]. Although this Φ X (L) is constructed from link diagrams, in §5 we later explain its topological meaning, together with a computation of H Q 2 (X) following from Eisermann [E1, E2].…”

“…We here note a relation between Π 2 (X) and quandle homology groups. As is known, H Q 2 (X) ∼ = Z/2 and H Q 3 (X) ∼ = Z/4 [CJKLS,Remark 6.10]. Namely, the summand Z/8 of Π 2 (X) is evaluated not by the quandle cohomology, but by the group cohomology H 3 gr (Q 8 ; Z/8).…”