Quivers for 3-manifolds: the correspondence, BPS states, and 3d $$ \mathcal{N} $$ = 2 theories

Abstract We introduce and explore the relation between quivers and 3-manifolds with the topology of the knot complement. This idea can be viewed as an adaptation of the knots-quivers correspondence to Gukov-Manolescu invariants of knot complements (also known as FK or $$ \hat{Z} $$ Z ̂ ). Apart from assigning quivers to complements of T(2,2p+1) torus knots, we study the physical interpretation in terms of the BPS spectrum and general structure of 3d $$ \mathcal{N} $$ N = 2 theories associated to both sides of the correspondence. We also make a step towards categorification by proposing a t-deformation of all objects mentioned above.

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Refined large N duality for knots

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216520410011 ◽

2020 ◽

pp. 2041001

Author(s):

Masaya Kameyama ◽

Satoshi Nawata

Keyword(s):

Moduli Spaces ◽

Bound States ◽

Simons Theory ◽

Global Symmetry ◽

Torus Knots ◽

Torus Knot ◽

Large N ◽

Bps Invariants ◽

Bps States ◽

Chern Simons

We formulate large [Formula: see text] duality of [Formula: see text] refined Chern–Simons theory with a torus knot/link in [Formula: see text]. By studying refined BPS states in M-theory, we provide the explicit form of low-energy effective actions of Type IIA string theory with D4-branes on the [Formula: see text]-background. This form enables us to relate refined Chern–Simons invariants of a torus knot/link in [Formula: see text] to refined BPS invariants in the resolved conifold. Assuming that the extra [Formula: see text] global symmetry acts on BPS states trivially, the duality predicts graded dimensions of cohomology groups of moduli spaces of M2–M5 bound states associated to a torus knot/link in the resolved conifold. Thus, this formulation can be also interpreted as a positivity conjecture of refined Chern–Simons invariants of torus knots/links. We also discuss about an extension to non-torus knots.

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THE SIMPLEST HYPERBOLIC KNOTS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216599000195 ◽

1999 ◽

Vol 08 (03) ◽

pp. 279-297 ◽

Cited By ~ 19

Author(s):

PATRICK J. CALLAHAN ◽

JOHN C. DEAN ◽

JEFFREY R. WEEKS

Keyword(s):

Ad Hoc ◽

Crossing Number ◽

Hyperbolic Manifolds ◽

Torus Knots ◽

Initial Observation ◽

Knot Complements ◽

Standard Notion ◽

Dehn Fillings

While the crossing number is the standard notion of complexity for knots, the number of ideal tetrahedra required to construct the complement provides a natural alternative. We determine which hyperbolic manifolds with 6 or fewer ideal tetrahedra are knot complements, and explicitly describe the corresponding knots in the 3-sphere. Thus, these 72 knots are the simplest knots according to this notion of complexity. Many of these knots have the structure of twisted torus knots. The initial observation that led to the project was the abundance of knot complements with small Seifert-fibered Dehn fillings among the census manifolds. Since many of these knots have rather large crossing number they do not appear in the knot tables. Our methods, while ad hoc, yield some detailed information about the knot complements as well as the manifolds that arise from exceptional surgeries on these knots.

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Exceptional knot homology

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216516400034 ◽

2016 ◽

Vol 25 (03) ◽

pp. 1640003

Author(s):

Ross Elliot ◽

Sergei Gukov

Keyword(s):

Hecke Algebras ◽

Torus Knots ◽

Knot Invariants ◽

Exceptional Groups ◽

Double Affine Hecke Algebras ◽

Natural Home ◽

Knot Homology ◽

Affine Hecke Algebras ◽

Bps States ◽

Knots And Links

The goal of this paper is twofold. First, we find a natural home for the double affine Hecke algebras (DAHA) in the physics of BPS states. Second, we introduce new invariants of torus knots and links called hyperpolynomials that address the “problem of negative coefficients” often encountered in DAHA-based approaches to homological invariants of torus knots and links. Furthermore, from the physics of BPS states and the spectra of singularities associated with Landau–Ginzburg potentials, we also describe a rich structure of differentials that act on homological knot invariants for exceptional groups and uniquely determine the latter for torus knots.

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BPS states in the Ω-background and torus knots

Journal of High Energy Physics ◽

10.1007/jhep04(2014)164 ◽

2014 ◽

Vol 2014 (4) ◽

Cited By ~ 3

Author(s):

K. Bulycheva ◽

A. Gorsky

Keyword(s):

Torus Knots ◽

Bps States

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Hyperbolic knot complements without closed embedded totally geodesic surfaces

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700001476 ◽

2000 ◽

Vol 68 (3) ◽

pp. 379-386 ◽

Cited By ~ 4

Author(s):

Kazuhiro Ichihara ◽

Makoto Ozawa

Keyword(s):

Torus Knots ◽

Totally Geodesic ◽

Essential Surface ◽

Knot Complements ◽

Double Torus ◽

Geodesic Surfaces

AbstractIt is conjectured that a hyperbolic knot complement does not contain a closed embedded totally geodesic surface. In this paper, we show that there are no such surfaces in the complements of hyperbolic 3-bridge knots and double torus knots. Some topological criteria for a closed essential surface failing to be totally geodesic are given. Roughly speaking, sufficiently ‘complicated’ surfaces cannot be totally geodesic.

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Atmospheric Circulation Systems - Their Structure and Physical Interpretation

10.1016/s0074-6142(08)x6048-6 ◽

1969 ◽

Keyword(s):

Atmospheric Circulation ◽

Physical Interpretation

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Cloning of the Rat Tissue Factor cDNA and Promoter: Identification of a Serum-response Region

Thrombosis and Haemostasis ◽

10.1055/s-0038-1650646 ◽

1996 ◽

Vol 76 (05) ◽

pp. 697-702 ◽

Cited By ~ 10

Author(s):

Olivier Taby ◽

Claire-Lise Rosenfield ◽

Vladimir Bogdanov ◽

Yale Nemerson ◽

Mark B Taubman

Keyword(s):

Tissue Factor ◽

General Structure ◽

Amino Acid Sequences ◽

Proximal Promoter ◽

Striking Similarity ◽

Upstream Sequence ◽

Tf Gene ◽

Rat Tissue ◽

High Degree ◽

Transcriptional Elements

SummaryTissue factor (TF) initiates coagulation and its expression in vascular smooth muscle cells (VSMC) likely plays a role in the propagation of arterial thrombosis. We report cloning the cDNA and proximal promoter region of the rat TF gene. While maintaining the general structure and organization of the TF molecule, there is a surprising divergence (≈ 18%) between the derived amino acid sequences of the rat and mouse TF. In contrast, there is striking similarity (90%) in the 5’ untranslated regions. High levels of basal promoter activity were seen in rat VSMC with constructs containing 106 bp of sequence downstream from the putative transcription start site and 426 to 103 bp of upstream sequence. Deletion of the sequence from −103 to −79, containing a single SP1 site, removed virtually all of the basal and serum-induced activity. Removal of the NFkB site or two additional upstream SP1 sites had little effect on serum responsiveness. Removal of the 5’ untranslated region abolished most of the basal activity of the TF promoter, suggesting that its high degree of conservation may be due to the presence of transcriptional elements critical for TF expression in rodent VSMC.

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