Related papers: The $\mathfrak{sl}_N$ Symmetrically Large Coloured…

The Gukov-Manolescu series, denoted by $F_K$, is a conjectural invariant of knot complements that, in a sense, analytically continues the colored Jones polynomials. In this paper we use the large color $R$-matrix to study $F_K$ for some…

We construct the invariant $F_K^{\mathfrak{sl}_3}\in\mathbb{Z}[q,q^{-1}][[x,y]]$ for any positive braid knot $K$, whose existence was conjectured by Park, building on earlier work of Gukov--Manolescu. The main step in our work extends a…

We introduce a three variable series invariant $F_K (y,z,q)$ for plumbed knot complements associated with a Lie superalgebra $sl(2|1)$. The invariant is a generalization of the $sl(2|1)$-series invariant $\hat{Z}(q)$ for closed 3-manifolds…

We explore the topological significance of the Gukov-Manolescu knot series $F_K$. We show that the leading coefficient of $F_K$ is a monomial and express its exponent in terms of the Hopf invariant for all homogeneous braid knots, and for…

We study $q$-series-valued invariants of 3-manifolds that depend on the choice of a root system $G$. This is a natural generalization of the earlier works by Gukov-Pei-Putrov-Vafa [arXiv:1701.06567] and Gukov-Manolescu [arXiv:1904.06057]…

Let k be a field of characteristic not two or three, let $\mathfrak{g}$ be a finite-dimensional colour Lie algebra and let V be a finite-dimensional representation of $\mathfrak{g}$. In this article we give various ways of constructing a…

We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show that these invariants coincide with previous invariants defined by Khovanov for sl_2 and sl_3 and by Mazorchuk-Stroppel…

S. Gukov and C. Manolescu conjectured that the Melvin-Morton-Rozansky expansion of the colored Jones polynomials can be re-summed into a two-variable series $F_K(x,q)$, which is the knot complement version of the 3-manifold invariant…

Quantum knot invariants (like colored HOMFLY-PT or Kauffman polynomials) are a distinguished class of non-perturbative topological invariants. Any known way to construct them (via Chern-Simons theory or quantum R-matrix) starts with a…

This article introduces a natural extension of colouring numbers of knots, called colouring polynomials, and studies their relationship to Yang-Baxter invariants and quandle 2-cocycle invariants. For a knot K in the 3-sphere let \pi_K be…

We define a limiting $\mathfrak{sl}_N$ Khovanov-Rozansky homology for semi-infinite positive multi-colored braids, and we show that this limiting homology categorifies a highest-weight projector for a large class of such braids. This…

We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show that these invariants coincide with previous invariants defined by Khovanov for sl(2) and sl(3) and by…

Knot polynomials colored with symmetric representations of $SL_q(N)$ satisfy difference equations as functions of representation parameter, which look like quantization of classical ${\cal A}$-polynomials. However, they are quite difficult…

We present a novel symmetry of the colored HOMFLY polynomial. It relates pairs of polynomials colored by different representations at specific values of $N$ and generalizes the previously known "tug-the-hook" symmetry of the colored…

We generalize the Morton-Franks-Williams inequality to the colored $\mathfrak{sl}(N)$ link homology defined in arXiv:0907.0695, which gives infinitely many new bounds for the braid index and the self linking number. A key ingredient of our…

We present a new conjectural symmetry of the colored Alexander polynomial, that is the specialization of the quantum $\mathfrak{sl}_N$ invariant widely known as the colored HOMFLY-PT polynomial. We provide arguments in support of the…

Building on Seidel-Solomon's fundamental work, we define the notion of a $\mathfrak{g}$-equivariant Lagrangian brane in an exact symplectic manifold $M$ where $\mathfrak{g} \subset SH^1(M)$ is a sub-Lie algebra of the symplectic cohomology…

Let $U_q(\hat{\cal G})$ be a quantized affine Lie algebra. It is proven that the universal R-matrix $R$ of $U_q(\hat{\cal G})$ satisfies the celebrated conjugation relation $R^\dagger=TR$ with $T$ the usual twist map. As applications, braid…

We define and study a family of link invariants $\mathit{HFK}_{n}(L)$. Although these homology theories are defined using holomorphic disc counts, they share many properties with $sl_{n}$ homology. Using these theories, we give a framework…

We present a ``reasonable'' statement about Lie algebras that is equivalent to the Four Color Theorem. The notions appearing in the statement also appear in the theory of finite-type invariants of knots (Vassiliev invariants) and…