The reduction of quantum invariants of 4-thickenings

Bobtcheva, Ivelina; Quinn, Frank

doi:10.4064/fm188-0-2

Cited by 2 publications

(4 citation statements)

References 20 publications

(45 reference statements)

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“…For type (3,1), let F = {−1, 0, 1} with the usual involution −(−1) = 1, −0 = 0. We assume for simplification that w(0) = 1 and | 0 0 0 0 0 0 | = 1.…”

Section: Examplesmentioning

confidence: 99%

“…We assume for simplification that w(0) = 1 and | 0 0 0 0 0 0 | = 1. Then, we have 41 equivalence classes of 6j-symbols and one remaining colour weight w (1). We obtain 1661 generators for the Turaev-Viro ideal, and after two days Singular succeeds with finding a Gröbner base with respect to some degree reverse lexicographic order formed by 1297 polynomials.…”

Section: Examplesmentioning

confidence: 99%

“…But it is a good news if one aims to construct invariants that potentially detect counter-examples of the Andrews-Curtis conjecture. Namely, by a result of Bobtcheva and Quinn [1], an invariant for Andrews-Curtis moves descending from a multiplicative invariant of 4-thickenings of special 2-polyhedra only depends on homology if the Euler characteristic of the 2-complex under consideration is at least 1. But a non-multiplicative ideal Turaev-Viro invariant for Andrews-Curtis moves [5] is potentially more useful.…”

Section: Propositionmentioning

confidence: 99%

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Ideal Turaev–Viro invariants

King¹

2007

Topology and its Applications

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A Turaev-Viro invariant is a state sum, i.e., a polynomial that can be read off from a special spine or a triangulation of a compact 3-manifold. If the polynomial is evaluated at the solution of a certain system of polynomial equations (Biedenharn-Elliott equations) then the result is a homeomorphism invariant of the manifold ("numerical Turaev-Viro invariant"). The equation system defines an ideal, and actually the coset of the polynomial with respect to that ideal is a homeomorphism invariant as well ("ideal Turaev-Viro invariant").It is clear that ideal Turaev-Viro invariants are at least as strong as numerical Turaev-Viro invariants, and we show that there is reason to expect that they are strictly stronger. They offer a more unified approach, since many numerical Turaev-Viro invariants can be captured in a singly ideal Turaev-Viro invariant. Using computer algebra, we obtain computational results on some examples of ideal Turaev-Viro invariants.

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“…For type (3,1), let F = {−1, 0, 1} with the usual involution −(−1) = 1, −0 = 0. We assume for simplification that w(0) = 1 and | 0 0 0 0 0 0 | = 1.…”

Section: Examplesmentioning

confidence: 99%

Section: Examplesmentioning

confidence: 99%

Section: Propositionmentioning

confidence: 99%

See 1 more Smart Citation

Ideal Turaev–Viro invariants

King¹

2007

Topology and its Applications

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“…of the category CW 1+1 of cobordisms of relative 2-dimensional CW-complexes modulo 2-equivalence. This last category was first defined in [14] and the corresponding (1+1)-TQFT's were studied in [2,4,12].…”

Section: Introductionmentioning

confidence: 99%

HKR-type invariants of 4–thickenings of 2–dimensional CW complexes

Bobtcheva

Messia

2003

Algebr. Geom. Topol.

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The HKR (Hennings-Kauffman-Radford) framework is used to construct invariants of 4-thickenings of 2-dimensional CW complexes under 2-deformations (1-and 2-handle slides and creations and cancellations of 1-2 handle pairs). The input of the invariant is a finite dimensional unimodular ribbon Hopf algebra A and an element in a quotient of its center, which determines a trace function on A. We study the subset T 4 of trace elements which define invariants of 4-thickenings under 2-deformations. In T 4 two subsets are identified : T 3 ⊂ T 4 , which produces invariants of 4-thickenings normalizable to invariants of the boundary, and T 2 ⊂ T 4 , which produces invariants of 4-thickenings depending only on the 2-dimensional spine and the second Whitney number of the 4-thickening. The case of the quantum sl(2) is studied in details. We conjecture that sl(2) leads to four HKR-type invariants and describe the corresponding trace elements. Moreover, the fusion algebra of the semisimple quotient of the category of representations of the quantum sl(2) is identified as a subalgebra of a quotient of its center.

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The reduction of quantum invariants of 4-thickenings

Cited by 2 publications

References 20 publications

Ideal Turaev–Viro invariants

Ideal Turaev–Viro invariants

HKR-type invariants of 4–thickenings of 2–dimensional CW complexes

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