Knots-quivers correspondence

Abstract

We introduce and explore the relation between knot invariants and quiver representation theory, which follows from the identification of quiver quantum mechanics in D-brane systems representing knots. We identify various structural properties of quivers associated to knots, and identify such quivers explicitly in many examples, including some infinite families of knots, all knots up to 6 crossings, and some knots with thick homology. Moreover, based on these properties, we derive previously unknown expressions for colored HOMFLY-PT polynomials and superpolynomials for various knots. For all knots, for which we identify the corresponding quivers, the LMOV conjecture for all symmetric representations (i.e. integrality of relevant BPS numbers) is automatically proved.

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

“…The knots-quivers correspondence [1,2] is a conjecture that for each knot K there exist a quiver Q and integers…”

“…\K implicitly depend on the gauge group and results of [3] correspond to the simplest nontrivial case of SU (2). General case is analysed in [45], but it is often very involved from computational point of view.…”

“…In recent years many interesting relations between quantum field theory, string theory, low-dimensional topology, and representation theory have been studied. In this paper we focus on the knots-quivers correspondence [1,2] and the Gukov-Manolescu invariants of knot complements [3]. (We will abbreviate them to GM invariants and denote F K -often this symbol is also used as their name.…”

“…In consequence, by assigning a quiver to the knot, we automatically prove the integrality of LMOV invariants, which is the statement of the LMOV conjecture [4][5][6]. This was done for all knots up to 6 crossings and infinite families of T (2,2p+1) JHEP09(2020)075 torus knots and twist knots in [2]. More systematic approach in terms of tangles provided a proof of the knots-quivers correspondence for all two-bridge knots [10] and recently for all arborescent knots [11], however finding a general proof remains to be an open problem.…”

“…Moreover, since HOMFLY-PT polynomials admit t-deformation in terms of superpolynomials [48], we can follow [43] and conjecture results for a, t-deformed GM invariants. We also study the physical intepretation of the new correspondence (for clarity we will refer to the knots-quivers correspondence of [1,2] as the standard one), generalising the results of [12,13]. It includes an exploration of the duality between 3d N = 2 theories associated to the knot complement and the quiver, as well as a study of their BPS states, which allows us to introduce knot complement analogs of LMOV invariants via DT invariants.…”