State-sum invariants of knotted curves and surfaces from quandle cohomology

Carter, J.; Jelsovsky, Daniel; Kamada, Seiichi; Langford, Laurel; Saito, Masahico

doi:10.1090/s1079-6762-99-00073-6

Cited by 42 publications

(23 citation statements)

References 26 publications

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“…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 .…”

Section: Amentioning

confidence: 99%

“…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).…”

Section: Put Itmentioning

confidence: 99%

“…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).…”

Section: Quandle Invariants Are Specialized Colouring Polynomialsmentioning

confidence: 99%

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Knot colouring polynomials

Eisermann

2007

Pacific J. Math.

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We introduce a natural extension of the colouring numbers of knots, called colouring polynomials, and study their relationship to Yang-Baxter invariants and quandle 2-cocycle invariants.For a knot K in the 3-sphere, let π K be the fundamental group of the knot complement ‫ޓ‬ 3 K , and let m K , l K ∈ π K be a meridian-longitude pair. Given a finite group G and an element x ∈ G we consider the set of representations ρ : π K → G with ρ(m K ) = x and define the colouring polynomial as ρ ρ(l K ). The resulting invariant maps knots to the group ring ‫ޚ‬G. It is multiplicative with respect to connected sum and equivariant with respect to symmetry operations of knots. Examples are given to show that colouring polynomials distinguish knots for which other invariants fail, in particular they can distinguish knots from their mutants, obverses, inverses, or reverses.We prove that every quandle 2-cocycle state-sum invariant of knots is a specialization of some knot colouring polynomial. This provides a complete topological interpretation of these invariants in terms of the knot group and its peripheral system. Furthermore, we show that the colouring polynomial can be presented as a Yang-Baxter invariant, i.e. as the trace of some linear braid group representation. This entails that Yang-Baxter invariants can detect noninversible and nonreversible knots.

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“…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 .…”

Section: Amentioning

confidence: 99%

Section: Put Itmentioning

confidence: 99%

Section: Quandle Invariants Are Specialized Colouring Polynomialsmentioning

confidence: 99%

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Knot colouring polynomials

Eisermann

2007

Pacific J. Math.

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“…We have a 3-cocycle θ 5 ∈ Z (Z 5 , U (1)) satisfying the conditions (Coc1) and (Coc2) in Theorem 3•8, for example, as the following map. (2,2,4) • t 4 (2,4,2) • t (2,4,3) •t (3,1,2) • t 4 (3,1,3) • t (3,3,1) • t 4 (3,3,3) • t (3,4,2) • t 4…”

Section: •2 An Example Of Invariantsmentioning

confidence: 99%

“…Computing the state sum invariant I θ 5 with this 3-cocycle θ 5 for the lens spaces L(5, 1) and L (5,2), where their diagrams of labelled links are depicted in Figure 20 and 21 respectively, we have The Seifert manifold M = (S 2 ; 5, 5, . .…”

Section: •2 An Example Of Invariantsmentioning

confidence: 99%

Invariants of 3-manifolds derived from covering presentations

Hatakenaka

2010

Math. Proc. Camb. Phil. Soc.

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By a covering presentation of a 3-manifold, we mean a labelled link (i.e., a link with a monodromy representation), which presents the 3-manifold as the simple 4-fold covering space of the 3-sphere branched along the link with the given monodromy. It is known that two labelled links present a homeomorphic 3-manifold if and only if they are related by a finite sequence of some local moves. This paper presents a method for constructing topological invariants of 3-manifolds based on their covering presentations. The proof of the topological invariance is shown by verifying the invariance under the local moves. As an example of such invariants, we present the Dijkgraaf-Witten invariant of 3-manifolds.

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