TG-Hyperbolicity of virtual links

Adams, Colin; Eisenberg, Or; Greenberg, Jonah; Kapoor, Kabir; Liang, Zhen; O'Connor, Kate; Pacheco-Tallaj, Natalia; Wang, Yi

doi:10.1142/s0218216519500809

Cited by 13 publications

(21 citation statements)

References 16 publications

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“…The proof of Theorem 4.8 uses the fact that the modules V j satisfy the recurrence relation V j+1 = V 2 V j − V j−1 , the same recurrence relation defining the Chebyshev polynomials in (1).…”

Section: The Toroidal Colored Jones Polynomialmentioning

confidence: 99%

A Quantum Invariant of Links in $T^2 \times I$ with Volume Conjecture Behavior

Boninger¹

2020

Preprint

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We define a polynomial invariant J T n of links in the thickened torus. We call J T n the nth toroidal colored Jones polynomial, and show it satisfies many properties of the original colored Jones polynomial. Most significantly, J T n exhibits volume conjecture behavior. We prove the volume conjecture for the 2-by-2 square weave, and provide computational evidence for other links. We also give two equivalent constructions of J T n , one using operator invariants and another using the Kauffman bracket skein module of the torus. In the process we generalize the theory of operator invariants to links in T 2 × I, defining what we call a pseudo-operator invariant.

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Section: The Toroidal Colored Jones Polynomialmentioning

confidence: 99%

A Quantum Invariant of Links in $T^2 \times I$ with Volume Conjecture Behavior

Boninger¹

2020

Preprint

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“…See [4] for a more complete introduction to hyperbolicity for virtual knots. That paper includes a table of the volumes of the 116 nontrivial virtual knots of four or fewer classical crossings calculated via the computer program SnapPy [8], all of which, with the exception of the trefoil knot, turn out to be tg-hyperbolic.…”

Section: Introductionmentioning

confidence: 99%

TG-Hyperbolicity of Composition of Virtual Knots

Adams¹,

Simons²

2021

Preprint

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The composition of any two nontrivial classical knots is a satellite knot, and thus, by work of Thurston, is not hyperbolic. In this paper, we explore the composition of virtual knots, which are an extension of classical knots that generalize the idea of knots in S 3 to knots in S × I where S is a closed orientable surface. In particular, we prove that virtual knots behave very differently under composition. The composition of two hyperbolic virtual knots is hyperbolic. We then obtain strong lower bounds on the volume of the composition using information about the original knots.

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“…Similarly, a classical non-splittable link that does not contain an essential torus or annulus is hyperbolic. In [2], hyperbolic invariants are extended to the virtual category by utilizing the equivalence of virtual links to links in thickened surfaces. Definition 1.1.…”

Section: Introductionmentioning

confidence: 99%

“…Thus, we can define hyperbolic invariants of the original virtual link accordingly. See [2] for more on this, including a table of volumes of virtual knots of four or fewer classical crossings.…”

Section: Introductionmentioning

confidence: 99%

Turaev Hyperbolicity of Classical and Virtual Knots

Adams,

Eisenberg,

Greenberg

et al. 2019

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By work of W. Thurston, knots and links in the 3-sphere are known to either be torus links, or to contain an essential torus in their complement, or to be hyperbolic, in which case a unique hyperbolic volume can be calculated for their complement. We employ a construction of Turaev to associate a family of hyperbolic 3-manifolds of finite volume to any classical or virtual link, even if non-hyperbolic. These are in turn used to define the Turaev volume of a link, which is the minimal volume among all the hyperbolic 3-manifolds associated via this Turaev construction. In the case of a classical link, we can also define the classical Turaev volume, which is the minimal volume among all the hyperbolic 3-manifolds associated via this Turaev construction for the classical projections only. We then investigate these new invariants.

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TG-Hyperbolicity of virtual links

Cited by 13 publications

References 16 publications

A Quantum Invariant of Links in $T^2 \times I$ with Volume Conjecture Behavior

A Quantum Invariant of Links in $T^2 \times I$ with Volume Conjecture Behavior

TG-Hyperbolicity of Composition of Virtual Knots

Turaev Hyperbolicity of Classical and Virtual Knots

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