On The Virtual Cosmetic Surgery Conjecture

Boyle, Keegan

doi:10.48550/arxiv.1701.02361

Cited by 1 publication

(2 citation statements)

References 7 publications

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“…Remark 3.6. Equation (15) does not characterize the series Z a (q) uniquely. Indeed, consider Euler's pentagonal series…”

Section: Conventionsmentioning

confidence: 99%

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A two-variable series for knot complements

Gukov¹,

Manolescu²

2019

Preprint

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The physical 3d N = 2 theory T [Y ] was previously used to predict the existence of some 3-manifold invariants Za(q) that take the form of power series with integer coefficients, converging in the unit disk. Their radial limits at the roots of unity should recover the Witten-Reshetikhin-Turaev invariants. In this paper we discuss how, for complements of knots in S 3 , the analogue of the invariants Za(q) should be a two-variable series FK (x, q) obtained by parametric resurgence from the asymptotic expansion of the colored Jones polynomial. The terms in this series should satisfy a recurrence given by the quantum A-polynomial. Furthermore, there is a formula that relates FK (x, q) to the invariants Za(q) for Dehn surgeries on the knot. We provide explicit calculations of FK (x, q) in the case of knots given by negative definite plumbings with an unframed vertex, such as torus knots. We also find numerically the first terms in the series for the figure-eight knot, up to any desired order, and use this to understand Za(q) for some hyperbolic 3-manifolds.

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“…Remark 3.6. Equation (15) does not characterize the series Z a (q) uniquely. Indeed, consider Euler's pentagonal series…”

Section: Conventionsmentioning

confidence: 99%

“…Thus, to every Z a (q) we could add (q) ∞ (times any polynomial in q, if we prefer) and get a new series that has the same limits at the roots of unity. However, physics predicts that there is a particular series Z a (q), among all those that satisfy (15), which is the one that should be categorified; cf. Remark 3.7 below.…”

Section: Conventionsmentioning

confidence: 99%

A two-variable series for knot complements

Gukov¹,

Manolescu²

2019

Preprint

View full text Add to dashboard Cite

show abstract

On The Virtual Cosmetic Surgery Conjecture

Cited by 1 publication

References 7 publications

A two-variable series for knot complements

A two-variable series for knot complements

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