Lectures on Knot Homology and Quantum Curves

Gukov, Sergei; Saberi, Ingmar

doi:10.1142/9789814630627_0004

Cited by 3 publications

(3 citation statements)

References 45 publications

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“…The trefoil. The A-polynomial for the right-handed trefoil K = 3 r 1 can be read off, for example, from [31, Section 3.2] or [38,Example 6]. Conjugating it with x 1 2 − x − 1 2 , we find the recursion relation (158) for F K (x, q) in this case:…”

Section: Knotmentioning

confidence: 99%

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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Section: Knotmentioning

confidence: 99%

A two-variable series for knot complements

Gukov¹,

Manolescu²

2019

Preprint

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“…For example, for a fixed link and labelling, it is well-known that the sl N invariants stabilize for large N to a colored HOMFLY-PT invariant in two variables a and q, which contains generalizations of the Alexander polynomial and from which the sl N invariants can be recovered as one-variable specializations a = q N . The dependence of HOMFLY-PT invariants on the labelling is governed by recurrence relations [17], which are notoriously difficult to compute and are conjectured to have deep relationships to the A-polynomial, character varieties and possibly even knot contact homology, see [16,15,14,21] and references therein. However, as regards questions like the one about dependence of the invariant under variations of the input link, the RT invariants seem to carry too little structure to allow meaningful answers.…”

Section: Introductionmentioning

confidence: 99%

Exponential growth of colored HOMFLY-PT homology

Wedrich

2019

Advances in Mathematics

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We define reduced colored sl N link homologies and use deformation spectral sequences to characterize their dependence on color and rank. We then define reduced colored HOMFLY-PT homologies and prove that they arise as large N limits of sl N homologies. Together, these results allow proofs of many aspects of the physically conjectured structure of the family of type A link homologies. In particular, we verify a conjecture of Gorsky, Gukov and Stošić about the growth of colored HOMFLY-PT homologies. arXiv:1602.02769v1 [math.GT] 8 Feb 2016 Theorem 2. . Let K be a labelled knot. For sufficiently large N , there are isomorphisms i+2N j=I h−j=Jbetween a grading-collapsed version of reduced colored HOMFLY-PT homology and reduced colored sl N homology. See Theorem 3.46.Theorem 2 relies on the fact that the reduced colored HOMFLY-PT homology is finite-dimensional for knots, see Proposition 3.45. On the contrary, the original unreduced colored HOMFLY-PT homology of Mackaay-Stošić-Vaz [36] and Webster-Williamson [53] is always infinite-dimensional. Nevertheless, we find an analogue of Theorem 1, which relates it to unreduced colored sl N homologies and -more generally -their Σ-deformed versions, which have been defined by Wu in [58] and further studied by Rose and the author in [49].Theorem 3. Let L be a labelled link. There is a spectral sequencefrom unreduced colored HOMFLY-PT homology to unreduced, Σ-deformed colored sl N homology. See Theorem 3.22.1.2. The physical structure. The existence of reduced colored sl N and HOMFLY-PT homologies and the theorems describing their relationships, although new, will not come as a surprise to the expert. In fact, they are at the very centre of current interest in the study of link homologies from the prespective of theoretical physics. Since Gukov-Schwarz-Vafa's physical interpretation of Khovanov-Rozansky homology as a space of BPS states [22], there has been significant cross-fertilization between the mathematical and physical sides of this field. A particularly interesting outcome is a package of conjectures about the structure of the family of colored sl N and HOMFLY-PT homologies, which has been developed over a decade in a series of papers by Dunfield-Gukov-Rasmussen [12], Gukov-Walcher [24], Gukov-Stošić [23], Gorsky-Gukov-Stošić [18] and most recently Gukov-Nawata-Saberi-Stošić-Su lkowski [20]. Thanks to recent advances in understanding the higher representation theory underlying link homologies (in particular categorical skew Howe duality), which led to the convergence of several different approaches to link homologies [7,42,49,37], we are now in a position to prove a significant part of the physically conjectured structure. In the following theorems, K k denotes a knot labelled with the k th exterior power of the vector representation.Theorem 4. There is a spectral sequencewhich preserves the homological grading. See Corollary 3.47.This implies that the colored HOMFLY-PT homologies of a knot grow at least exponentially in color, as predicted in [18]. A related result has ...

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“…Knot Homology and Knot Invariants [9] Lets consider the category of finite-dimensional vector spaces and linear maps. We naturally associated a number to each object in this category, the dimension of that vector space.…”

Section: Chaptermentioning

confidence: 99%

Knot Invariants and Topological Quantum Field Theory

Akhtar¹

2021

Preprint

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An elementary introduction to knot theory and its link to quantum field theory is presented with an intention to provide details of some basic calculations in the subject, which are not easily found in texts. Study of Chern-Simons theory with gauge group G, along with the Wilson lines carrying some representation is explained in generality, and a vital calculation of the Chern-Simons propagator is done. Explicit calculation for U (1) Chern-Simons theory is presented, which leads to the topological invariants, and finally to knot invariants. Further, using this result along with the Gauss linking number formula, the expectation value of Wilson loops are calculated. Colored knot invariants are also discussed along with more advanced knot invariants which are obtained using Homology theory, i.e., categorification of Jones and HOMFLY polynomials. Various knot invariants for SU (N ) gauge group are also introduced, along with a brief introduction to A-polynomials and super A-polynomials. Recent developments in the field are explored, and we discuss a conjectured formula for colored superpolynomials, closed-form expression for HOMFLY polynomials, and conjectured expression for 6j symbol for U q (sl N ) for multiplicity free case. Also, a MATHEMATICA program based on the conjectured formula had been developed, which can compute the 6j-symbols and the desired duality matrices which are needed to use the closed-form expression for HOMFLY polynomials.

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Lectures on Knot Homology and Quantum Curves

Cited by 3 publications

References 45 publications

A two-variable series for knot complements

A two-variable series for knot complements

Exponential growth of colored HOMFLY-PT homology

Knot Invariants and Topological Quantum Field Theory

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