arXiv research

For each graph and each positive integer $n$, we define a chain complex whose graded Euler characteristic is equal to an appropriate $n$-specialization of the dichromatic polynomial. This also gives a categorification of $n$-specializations of the Tutte polynomial of graphs. Also, for each graph and integer $n\le 2$, w…

Study links and quivers, proving polynomial equality conjecture.

The taut polynomial equals a twisted Alexander polynomial.

Refines virtual link equality criterion for diagrams with one virtual crossing.

Study connects group invariants through outer automorphisms and polynomial relations.

Polynomials with distinct critical values have braid monodromy groups equal to braid groups.

We develop an invariant of knots that depends on a complex parameter t, describing a left ideal in the noncommutative torus. When the parameter is set equal to -1 we recover the A-polynomial of the knot. We relate the invariant to the colored Jones polynomials of the knot.

Alexander polynomial equals spanning tree count at t=1.

The interior polynomial is an 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 a par…

New bound on Jones polynomial for specific positive links.

New knots with specific properties have identical polynomial values.

Direct proof of Alexander polynomial scaling for L-shaped representations.

We define and calculate the HOMFLY polynomial for a specific type of quiver.

We describe a correspondence between augmentations and certain representations of the knot group. The correspondence makes the 2-variable augmentation polynomial into a generalization of the classical $A$-polynomial. It also associates to an augmentation a rank, which is bounded by the bridge number and shares its beha…

Computed formulas for curvature operators and Poincaré polynomials of symmetric spaces.

The volume conjecture and its generalization state that the series of certain evaluations of the colored Jones polynomials of a knot would grow exponentially and its growth rate would be related to the volume of a three-manifold obtained by Dehn surgery along the knot. In this paper, we show that for the figure-eight k…

Study shows link polynomial evaluations from Heegaard Floer theory.

Numerical discovery matches eta invariant on Berger spheres with conformal anomaly on round spheres.

Researchers prove a conjecture linking 1-loop invariants to torsion for fibered 3-manifolds.

Simple proof of knot genus theorem using Alexander polynomial.

Using the Fourier expansion of Markov traces for Ariki-Koike algebras over $\mathbb{Q}(q,u_{1},...,u_{e})$, we give a direct definition of the Alexander polynomials for mixed links. We observe that under the corresponding specialization of a Markov parameter, the Fourier coefficients of Markov traces take quite simple …

For any positive integer r, we exhibit a knot Kr with (20 $\times$ 2 r--1 + 1) crossings whose Jones polynomial V (Kr) is equal to 1 mod-ulo 2 r. Our construction rests on a certain 20-crossing tangle T 20 which is undetectable by the Kauffman bracket polynomial pair mod 2.

Jones polynomials have infinitely many roots of unity as zeros.

Developed algorithms to compute three polynomial invariants of veering triangulations.

The interior polynomial is an invariant of (signed) bipartite graphs, and the interior polynomial of a plane bipartite graph is equal to a part of the HOMFLY polynomial of a naturally associated link. The HOMFLY polynomial $P_L(v,z)$ is a famous link invariant with many known properties. For example, the HOMFLY polynom…

It is known that the minimal degree of the Jones polynomial of a positive knot is equal to its genus, and the minimal coefficient is 1. We extend this result to almost positive links and partly identify the 3 following coefficients for special types of positive links. We also give counterexamples to the Jones polynomia…

New skein theory for Links-Gould polynomial simplifies link evaluations.

New bounds on Jones polynomial positivity for specific links.

This paper defines girth for knots and links, linking it to Khovanov homology.

Study shows torsion homology growth vanishes for certain free-by-cyclic groups.

The Jones polynomial $V_{L}(t)$ for an oriented link $L$ is a one-variable Laurent polynomial link invariant discovered by Jones. For any integer $n\ge 3$, we show that: (1) the difference of Jones polynomials for two oriented links which are $C_{n}$-equivalent is divisible by $\left(t-1\right)^{n}\left(t^{2}+t+1\right…

For a knot $K$ in $S^3$, the $sl_2$-colored Jones function $J_K(n)$ is a sequence of Laurent polynomials in the variable $t$, which is known to satisfy non-trivial linear recurrence relations. The operator corresponding to the minimal linear recurrence relation is called the recurrence polynomial of $K$. The AJ conject…

The paper classifies pretzel links with 2 components and gives conditions for those with 3 or more.

New knot polynomials reveal patterns and mutations.

New bounds on HOMFLY polynomial for homogeneous links.

We show two results about the Conway potential function which is known as the normalized multivariable Alexander polynomial. We first show that the Conway potential function introduced by Kauffman in "Formal Knot Theory" is indeed a link invariant. Next we show that Kauffman's potential function equals Hartley's potent…

Study harmonic functions on submanifolds and their cones.

This paper consists of three parts. First, we generalize the Jaeger Formula to express the Kauffman-Vogel graph polynomial as a state sum of the Murakami-Ohtsuki-Yamada graph polynomial. Then, we demonstrate that reversing the orientation and the color of a MOY graph along a simple circuit does not change the sl(N) Mur…

In this article, we give an elementary construction of homological invariants of links presented by braid closures. The Euler characteristic of this complex is equal to quantum polynomial invariant of link.

Bounds on knot polynomials for Lie superalgebras of type I.

Study Betti and Hodge numbers of solvmanifolds from integer polynomials.

It has been known that any Alexander polynomial of a knot can be realized by a quasipositive knot. As a consequence, the Alexander polynomial cannot detect quasipositivity. In this paper we prove a similar result about Vassiliev invariants: for any oriented knot K and any natural number n there exists a quasipositive k…

C. Giller proposed an invariant of ribbon 2-knots in S^4 based on a type of skein relation for a projection to R^3. In certain cases, this invariant is equal to the Alexander polynomial for the 2-knot. Giller's invariant is, however, a symmetric polynomial -- which the Alexander polynomial of a 2-knot need not be. Afte…

New lower bound for knot genus using Links-Gould invariant.

Developed a new homology theory for graph chromatic polynomials.

Polynomial-time algorithm learns causal graphs without parametric assumptions.

The nonzero level sets in $n$-dimensional flat affine space of a translationally homogeneous function are improper affine spheres if and only if the Hessian determinant of the function is equal to a nonzero constant multiple of the $n$th power of the function. The exponentials of the characteristic polynomials of certa…

R.M. Kashaev conjectured that the asymptotic behavior of his link invariant, which equals the colored Jones polynomial evaluated at a root of unity, determines the hyperbolic volume of any hyperbolic link complement. We observe numerically that for knots $6_3$, $8_9$ and $8_{20}$ and for the Whitehead link, the colored…