A closed formula for the evaluation of foams

Robert, Louis-Hadrien; Wagner, Emmanuel

doi:10.4171/qt/139

Cited by 31 publications

(58 citation statements)

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“…(c) Foams are suitably decorated 2-dimensional CW-complexes, defined abstractly or embedded in R 3 . They originate and most prominently appear in the study of link homologies, see for example [Kho04], [EST17], [RW20] or [ETW18]. Using the universal construction from [BHMV95], they can easily modified, see e.g.…”

Section: Symbolmentioning

confidence: 99%

Monoidal categories, representation gap and cryptography

Khovanov¹,

Sitaraman²,

Tubbenhauer³

2022

Preprint

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The linear decomposition attack provides a serious obstacle to direct applications of noncommutative groups and monoids in cryptography. To overcome this issue we propose to look at monoids with only big representations, in the sense made precise in the paper, and undertake a systematic study of such monoids. One of our main tools is Green's theory of cells (Green's relations).A large supply of monoids is delivered by monoidal categories. We consider simple examples of monoidal categories of diagrammatic origin, including the Temperley-Lieb, the Brauer and partition categories, and discuss lower bounds for their representations.

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Section: Symbolmentioning

confidence: 99%

Monoidal categories, representation gap and cryptography

Khovanov¹,

Sitaraman²,

Tubbenhauer³

2022

Preprint

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“…The idea to extend the Robert-Wagner foam evaluation [RW20] to the case when F i (c) are not closed surfaces by capping their boundary circles with disks and taking the Euler characteristic of the resulting surfaces F i (c) was proposed by Yakov Kononov [Kon19], who also pointed out that such closure constructions are used implicitly in the physics TQFT literature. The alternative is to use the Euler characteristic of F 1 (c), which may be an odd integer.…”

Section: An Evaluation Over R With Defect Linesmentioning

confidence: 99%

“…Remark. Equation (3.3) has α 2 − α 1 in the denominator, compared to X i − X j for i < j in [RW20]. This is done to make the 2-sphere with one dot evaluate to 1 rather than −1.…”

Section: An Evaluation Over R With Defect Linesmentioning

confidence: 99%

Link homology and Frobenius extensions II

Khovanov

Robert

2022

Fund. Math.

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The first two sections of the paper provide a convenient scheme and additional diagrammatics for working with Frobenius extensions responsible for key flavors of equivariant SL(2) link homology theories. The goal is to clarify some basic structures in the theory and propose a setup to work over sufficiently non-degenerate base rings. The third section works out two related SL(2) evaluations for seamed surfaces.

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“…In section 3.1, we briefly describe the version of slpNq link homology that we use. Its construction uses a cube of resolutions and Robert-Wagner's combinatorial evaluation of closed foams [RW20]. We refer to [Wan21b] for a detailed exposition of the construction, and for the definitions of slpNq MOY graphs and slpNq foams.…”

Section: Slpnq Link Homologymentioning

confidence: 99%

“…Khovanov homology is the N " 2 case of slpNq link homology [KR08], which can also be defined combinatorially [RW20], but is much less well-understood. Using a spectral sequence relating slpNq link homology and Khovanov homology [Wed19], some detection results for slpNq link homology can be obtained from the analogous detection results for Khovanov homology.…”

Section: Introductionmentioning

confidence: 99%