HOMFLY Research Articles

Forty years in mathematics: Personal perspective (Part I)

Conference like this in Dallas is a perfect occasion to give some reminiscences of my 40 years of work as a mathematician, and to muse a little about my mathematical past. It is a rare occasion to be a little sentimental. In this essay, I summarize and expand my birthday talk given at University of Texas at Dallas conference. There, I mostly covered the early years of my mathematical career ending with a detailed description about the smallest volume hyperbolic 3-manifold and the HOMFLYPT polynomial. Thus I seldom venture to the time after 1984 (for example to mention skein modules constructed in April 1987) and only occasionally after 1995 when I settled to work at George Washington University and have had several successful PhD students.

The second Yang–Baxter homology for the HOMFLYPT polynomial

In this paper, we adjust the Yang–Baxter operators constructed by Jones for the HOMFLYPT polynomial. Then we compute the second homology for this family of Yang–Baxter operators. It has the potential to yield 2-cocycle invariants for links.

Chern-Simons perturbative series revisited

A group-theoretical structure in a perturbative expansion of the Wilson loops in the 3d Chern-Simons theory with SU(N) gauge group is studied in symmetric approach. A special basis in the center of the universal enveloping algebra ZU(slN) is introduced. This basis allows one to present group factors in an arbitrary irreducible finite-dimensional representation. Developed methods have wide applications, the most straightforward and evident ones are mentioned. Namely, Vassiliev invariants of higher orders are computed, a conjecture about existence of new symmetries of the colored HOMFLY polynomials is stated, and the recently discovered tug-the-hook symmetry of the colored HOMFLY polynomial is proved.

A full-twisting formula for the HOMFLY polynomial

We introduce a certain class of link diagrams, which includes all closed braid diagrams. We show a generalized version of K\'alm\'an's full-twist formula for the HOMFLY polynomial in the class.

Harer-Zagier formulas for knot matrix models

Knot matrix models are defined so that the averages of characters are equal to knot polynomials. From this definition one can extract single trace averages and generation functions for them in the group rank – which generalize the celebrated Harer-Zagier formulas for Hermitian matrix model. We describe the outcome of this program for HOMFLY-PT polynomials of various knots. In particular, we claim that the Harer-Zagier formulas for torus knots factorize nicely, but this does not happen for other knots. This fact is mysteriously parallel to existence of explicit β=1 eigenvalue model construction for torus knots only, and can be responsible for problems with construction of a similar model for other knots.

Quantum Racah matrices and 3-strand braids in representation [3,3

This paper is devoted to the advance in the project of systematic description of colored knot polynomials started in [35] – explicit calculation of the inclusive Racah matrices for representation R=[3,3]. This is made possible by a powerful technique which we suggest in this paper – the use of highest weight method in the basis of Gelfand-Tseitlin tables. Our result allows one to evaluate and investigate [3,3]-colored polynomials for arbitrary 3-strand knots, and this confirms many previous conjectures on various factorizations, universality, and differential expansions. Furthermore, with the help of a method developed in [45] we manage to calculate exclusive Racah matrices in R=[3,3]. Our results confirm a calculation of these matrices in [51], which was based on the conjecture of explicit form of differential expansion for twist knots. Explicit answers for Racah matrices and [3,3]-colored polynomials for 3-strand knots up to 10 crossings are available at [1]. With the help of our results for inclusive and exclusive Racah matrices, it is possible to compute [3,3]-colored HOMFLY-PT polynomial of any link for the so-called one-looped family links, which are obtained from arborescent links by adding one loop.

A Novel Symmetry of Colored HOMFLY Polynomials Coming from $$\mathfrak {sl}(N|M)$$ Superalgebras

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 Alexander polynomial. As we show, the symmetry has a superalgebra origin, which we discuss qualitatively. Our main focus are the constraints that such a property imposes on the general group-theoretical structure, namely the $\mathfrak{sl}(N)$ weight system, arising in the perturbative expansion of the invariant. Finally, we demonstrate its tight relation to the eigenvalue conjecture.

Quantum Invariants of Knotoids

In this paper, we construct quantum invariants for knotoid diagrams in $\mathbb{R}^2$. The diagrams are arranged with respect to a given direction in the plane ({\it Morse knotoids}). A Morse knotoid diagram can be decomposed into basic elementary diagrams each of which is associated to a matrix that yields solutions of the quantum Yang-Baxter equation. We recover the bracket polynomial, and define the rotational bracket polynomial, the binary bracket polynomial, the Alexander polynomial, the generalized Alexander polynomial and an infinity of specializations of the Homflypt polynomial for Morse knotoids via quantum state sum models.

A diagrammatic approach for determining the braid index of alternating links

This paper concerns the braid index of an alternating link. It is well known that the braid index of any link equals the minimum number of Seifert circles among all link diagrams representing it. For an alternating link represented by a reduced alternating diagram [Formula: see text], it is known that [Formula: see text], the number of Seifert circles in [Formula: see text], equals the braid index [Formula: see text] of [Formula: see text] if [Formula: see text] contains no lone crossings, where a crossing in [Formula: see text] is called a lone crossing if it is the only crossing between two Seifert circles in [Formula: see text]. If [Formula: see text] contains lone crossings, then one can reduce the number of Seifert circles in [Formula: see text] using link-type preserving moves such as the Murasugi–Przytycki operation. Let [Formula: see text] be the maximum number of Seifert circles in [Formula: see text] that can be reduced, then we have [Formula: see text]. On the other hand, if the [Formula: see text]-span of the HOMFLY polynomial of [Formula: see text] satisfies the equality [Formula: see text]-span[Formula: see text], then the Morton–Frank–Williams (MFW) inequality [Formula: see text]-span[Formula: see text] leads us to the simple braid index formula [Formula: see text]. In this paper, we derive explicit formulas for many alternating links based on minimum projections of these links by establishing the equality [Formula: see text]-span[Formula: see text]. Our methods depend on the structures of the link diagrams under consideration and our results lead to explicit braid index formulas that are applicable to a very large class of links, a proper subset of which contains all two bridge links, all alternating pretzel links, and more generally all alternating Montesinos links. The derived braid index formula for an alternating Montesinos link is a function whose inputs are the signs of the crossings in the rational tangles of the Montesinos link. Finally, by applying the now proven Jones Conjecture on the writhe of minimum braids of a link, our results also allow us to obtain explicit formulas of the writhe of minimum braids for the links discussed in this paper from the minimum projections of these links.

Interplay between symmetries of quantum 6j-symbols and the eigenvalue hypothesis

The eigenvalue hypothesis claims that any quantum Racah matrix for finite-dimensional representations of $U_q(sl_N)$ is uniquely determined by eigenvalues of the corresponding quantum $\cal{R}$-matrices. If this hypothesis turns out to be true, then it will significantly simplify the computation of Racah matrices. Also due to this hypothesis various interesting properties of colored HOMFLY-PT polynomials will be proved. In addition, it allows one to discover new symmetries of the quantum 6-j symbols, about which almost nothing is known for $N>2$, with the exception of the tetrahedral symmetries, complex conjugation and transformation $q \longleftrightarrow q^{-1}$. In this paper we prove the eigenvalue hypothesis in $U_q(sl_2)$ case and show that it is equivalent to 6-j symbol symmetries (the Regge symmetry and two argument permutations). Then we apply the eigenvalue hypothesis to inclusive Racah matrices with 3 symmetric incoming representations of $U_q(sl_N)$ and an arbitrary outcoming one. It gives us 8 new additional symmetries that are not tetrahedral ones. Finally, we apply the eigenvalue hypothesis to exclusive Racah matrices with symmetric representations and obtain 4 tetrahedral symmetries.

Recurrence Relations for Knot Polynomials of Twist Knots

This paper gives HOMFLY polynomials and Kauffman polynomials L and F of twist knots as recurrence relations, respectively, and also provides some recursive properties of them.

Perturbative analysis of the colored Alexander polynomial and KP soliton τ-functions

In this paper we study the group theoretic structures of colored HOMFLY polynomials in a specific limit. The group structures arise in the perturbative expansion of SU(N) Chern-Simons Wilson loops, while the limit is N→0. The result of the paper is twofold. First, we explain the emergence of Kadomsev-Petviashvily (KP) τ-functions. This result is an extension of what we did in [1], where a symbolic correspondence between KP equations and group factors was established. In this paper we prove that integrability of the colored Alexander polynomial is due to it's relation to soliton τ-functions. Mainly, the colored Alexander polynomial is embedded in the action of the KP generating function on the soliton τ-function. Secondly, we use this correspondence to provide a rather simple combinatoric description of the group factors in term of Young diagrams, which is otherwise described in terms of chord diagrams, where no simple description is known. This is a first step providing an explicit description of the group theoretic data of Wilson loops, which would effectively reduce them to a purely topological quantity, mainly to a collection of Vassiliev invariants.

A New Symmetry of the Colored Alexander Polynomial

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 existence of the symmetry by studying the loop expansion and the character expansion of the colored HOMFLY-PT polynomial. We study the constraints this symmetry imposes on the group theoretic structure of the loop expansion and provide solutions to those constraints. The symmetry is a powerful tool for research on polynomial knot invariants and in the end we suggest several possible applications of the symmetry.

一类链环的HOMFLY多项式

一类链环的HOMFLY多项式

Vassiliev knot invariants derived from cable Γ-polynomials

For coprime integers p(>0) and q, the (p,q)-cable Γ-polynomial of a knot K is the Γ-polynomial of the (p,q)-cable knot of K, where the Γ-polynomial is the common zeroth coefficient polynomial of the HOMFLYPT and Kauffman polynomials. In this paper, we give some results on Vassiliev knot invariants derived from the cable Γ-polynomials. In particular, we show that all Vassiliev knot invariants of order ≤4 are determined by the cable Γ-polynomials.

Revisiting the Melvin-Morton-Rozansky expansion, or there and back again

Alexander polynomial arises in the leading term of a semi-classical Melvin-Morton-Rozansky expansion of colored knot polynomials. In this work, following the opposite direction, we propose how to reconstruct colored HOMFLY-PT polynomials, superpolynomials, and newly introduced hat{Z} invariants for some knot complements, from an appropriate rewriting, quantization and deformation of Alexander polynomial. Along this route we rederive conjectural expressions for the above mentioned invariants for various knots obtained recently, thereby proving their consistency with the Melvin-Morton-Rozansky theorem, and derive new formulae for colored superpolynomials unknown before. For a given knot, depending on certain choices, our reconstruction leads to equivalent expressions, which are either cyclotomic, or encode certain features of HOMFLY-PT homology and the knots-quivers correspondence.

Interior polynomial for signed bipartite graphs and the HOMFLY polynomial

The interior polynomial is a Tutte-type invariant of bipartite graphs, and a part of the HOMFLY polynomial of a special alternating link coincides with the interior polynomial of the Seifert graph of the link. We extend the interior polynomial to signed bipartite graphs, and we show that, in the planar case, it is equal to the maximal [Formula: see text]-degree part of the HOMFLY polynomial of a naturally associated link. Note that the latter can be any oriented link. This result fits into a program aimed at deriving the HOMFLY polynomial from Floer homology. We also establish some other, more basic properties of the signed interior polynomial. For example, the HOMFLY polynomial of the mirror image of [Formula: see text] is given by [Formula: see text]. This implies a mirroring formula for the signed interior polynomial in the planar case. We prove that the same property holds for any bipartite graph and the same graph with all signs reversed. The proof relies on Ehrhart reciprocity applied to the so-called root polytope. We also establish formulas for the signed interior polynomial inspired by the knot theoretical notions of flyping and mutation. This leads to new identities for the original unsigned interior polynomial.

Distinguishing Mutant knots

Knot theory is actively studied both by physicists and mathematicians as it provides a connecting centerpiece for many physical and mathematical theories. One of the challenging problems in knot theory is distinguishing mutant knots. Mutant knots are not distinguished by colored HOMFLY-PT polynomials for knots colored by either symmetric and or antisymmetric representations of SU(N). Some of the mutant knots can be distinguished by the simplest non-symmetric representation [2,1]. However there is a class of mutant knots which require more complex representations like [4,2]. In this paper we calculate polynomials and differences for the mutant knot polynomials in representations [3,1] and [4,2] and study their properties.

Derivation of the HOMFLYPT knot polynomial via helicity and geometric quantization

In this note we propose a geometric quantization interpretation of the HOMFLYPT polynomial, taking inspiration from the Liu-Ricca hydrodynamical approach to the latter and building on the Besana-S. symplectic approach to framing via Brylinski’s manifold of mildly singular links.

Cyclotomic expansions of HOMFLY-PT colored by rectangular Young diagrams

We conjecture a closed-form expression of HOMFLY-PT invariants of double twist knots colored by rectangular Young diagrams where the twist is encoded in interpolation Macdonald polynomials. We also put forth a conjecture of cyclotomic expansions of HOMFLY-PT polynomials colored by rectangular Young diagrams for any knot.