Formula calculates MOY webs and link polynomials.
problem No specific problem stated; focuses on evaluation.
method Closed formula for exterior webs and link polynomials.
result Closed formula for evaluating MOY webs and link polynomials.
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48 results for “polynomial evaluations”
Study shows link polynomial evaluations from Heegaard Floer theory.
problem Link polynomial evaluations from Heegaard Floer theory.
method Definition of Euler characteristic for fractionally-graded complexes based on roots of unity.
result Equality of Alexander polynomial evaluations and polynomial evaluations at certain roots of unity.
New relations for Penrose polynomial at n=4 and n=3.
problem Evaluation of Penrose polynomial at specific values of n.
method New ribbon graph polynomial and extended relations for n=3 to all n.
result Proven new relations for Penrose polynomial at n=4 and extended relations for n=3.
New formulas for knot polynomial evaluations from covering spaces.
problem Evaluating knot polynomials uniquely from covering spaces.
method Using singular determinants and linking pairings.
result Explicit formulae for Jones and -polynomial evaluations.
Recently V. Krushkal and D. Renardy generalized the Tutte polynomial from graphs to cell complexes. We show that evaluating this polynomial at the origin gives the number of cellular spanning trees in the sense of A. Duval, C. Klivans, and J. Martin. Moreover, after a slight modification, the Tutte-Krushkal-Renardy pol…
Evaluating the Jones polynomial with tensor networkscond-mat.stat-mech
We introduce tensor network contraction algorithms for the evaluation of the Jones polynomial of arbitrary knots. The value of the Jones polynomial of a knot maps to the partition function of a -state Potts model defined as a planar graph with weighted edges that corresponds to the knot. For any integer , we cast…
The volume conjecture and its generalizations say that the colored Jones polynomial corresponding to the N-dimensional irreducible representation of sl(2;C) of a (hyperbolic) knot evaluated at exp(c/N) grows exponentially with respect to N if one fixes a complex number c near 2*Pi*I. On the other hand if the absolute v…
New skein theory for Links-Gould polynomial simplifies link evaluations.
problem Computing Links-Gould polynomial for oriented links.
method Developed a cubic braid-type skein theory.
result Skein theory can evaluate any oriented link.
Murakami-Ohtsuki-Yamada introduced an evaluation of certain oriented planar trivalent graphs with colored edges. This evaluation plays a key role in the evaluation of the colored HOMFLY polynomial of a link in 3-space and its Khovanov-Rozansky categorification. Our goal is is to give a generating series formula for the…
We give a purely combinatorial formula for evaluating closed decorated foams. Our evaluation gives an integral polynomial and is directly connected to an integral equivariant version of the link homology categorifying the link polynomial. We also provide connections to the equivarian…
Study on colored Jones polynomial of figure-eight knot for complex parameters.
problem Asymptotic behavior of colored Jones polynomial for figure-eight knot.
method Analyzing the asymptotic growth rate of the polynomial for complex parameters with small imaginary part.
result Growth rate of polynomial is related to the Chern-Simons invariant for large real part of the parameter and to the reciprocal of Alexander polynomial for small real part.
Study shows quantum modularity in figure-eight knot's colored Jones polynomial.
problem Asymptotic behavior of colored Jones polynomial of figure-eight knot.
method Analyzing polynomial evaluated at specific points and showing asymptotic equivalence.
result Quantum modularity demonstrated in the figure-eight knot's colored Jones polynomial.
Researchers compute and predict knot volumes using colored Jones polynomials.
problem Computing and predicting volumes of hyperbolic knots.
method Vertex model approach, neural network training, polynomial evaluations.
result 3-colored Jones polynomials predict knot volumes with high accuracy.
We establish a relation between the trace evaluation in SO(3) topological quantum field theory and evaluations of a topological Tutte polynomial. As an application, a generalization of the Tutte golden identity is proved for graphs on the torus.
Unified ADO and colored Jones polynomials for knots.
problem Determining ADO polynomials from colored Jones polynomials.
method Constructing a two-variable knot invariant using completions of rings and algebra.
result Unified invariant maps colored Jones polynomials to ADO polynomials.
We analyze relationships between quantum computation and a family of generalizations of the Jones polynomial. Extending recent work by Aharonov et al., we give efficient quantum circuits for implementing the unitary Jones-Wenzl representations of the braid group. We use these to provide new quantum algorithms for appro…
Alexander polynomial equals spanning tree count at t=1.
problem Alexander polynomial for spatial graphs.
method Combinatorial constructions generalized to weighted graphs.
result Value of Alexander polynomial at t=1 equals weighted spanning tree count.
Qazaqzeh and Chbili showed that for any quasi-alternating link, the degree of -polynomial is less than its determinant. We give a refinement of their evaluation.
We illustrate from the viewpoint of braiding operations on WZNW conformal blocks how colored HOMFLY polynomials with multiplicity structure can detect mutations. As an example, we explicitly evaluate the (2,1)-colored HOMFLY polynomials that distinguish a famous mutant pair, Kinoshita-Terasaka and Conway knot.
New knot polynomials yield simple results modulo primes.
problem Understanding knot polynomials modulo primes.
method Constructing knots with specific properties.
result All polynomials modulo with bounded -span are realizable by knots with bounded braid index.
This paper presents an algorithm to construct a weighted adjacency matrix of a plane bipartite graph obtained from a pretzel knot diagram. The determinant of this matrix after evaluation is shown to be the Jones polynomial of the pretzel knot by way of perfect matchings (or dimers) of this graph. The weights are Tutte'…
Direct proof of one-hook scaling property for Alexander polynomial from Reshetikhin-Turaev formalismmath-ph
Direct proof of Alexander polynomial scaling for L-shaped representations.
problem Proving scaling property of Alexander polynomials for specific representations.
method Direct use of Reshetikhin-Turaev formalism to compute R-matrices.
result Normalized Alexander polynomial for one-hook representations scales with .
We show that the Kauffman bracket of a checkerboard colorable virtual link is an evaluation of the Bollobás-Riordan polynomial of a ribbon graph associated with . This result generalizes Thistlethwaite's celebrated theorem relating the Kauffman bracket with the Tutte polynomial of planar graphs.
It is known that evaluating a certain approximation to the Jones polynomial for the plat closure of a braid is a BQP-complete problem. That is, this problem exactly captures the power of the quantum circuit model. The one clean qubit model is a model of quantum computation in which all but one qubit starts in the maxim…
The volume conjecture for surface diffeomorphisms connects volumes to polynomial evaluations.
problem Connecting surface volumes to polynomial evaluations of quantum invariants.
method Developing combinatorial and algebraic techniques to compute isomorphisms between representations.
result Numerical evidence supports a conjecture linking surface volumes to polynomial evaluations.
F. Jaeger presented the two-variable Kauffman polynomial of an unoriented link L as a weighted sum of HOMFLY-PT polynomials of oriented links associated with L. Murakami, Ohtsuki and Yamada (MOY) used planar graphs and a recursive evaluation of these graphs to construct a state model for the sl(n)-link invariant (a one…
The Jones polynomial and functions of positive type on the oriented Jones-Thompson groups and math.GR
The pioneering work of Jones and Kauffman unveiled a fruitful relationship between statistical mechanics and knot theory. Recently, Jones introduced two subgroups and of the Thompson groups and , respectively, together with a procedure that associates an oriented link diagram to any element o…
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…
In this paper, we study the asymptotic behavior of the colored Jones polynomials evaluated at roots of unity for a special class of knots. We show that certain limit is zero as predicted by the volume conjecture.
We define two new invariants for tied links. One of them can be thought as an extension of the Kauffman polynomial and the other one as an extension of the Jones polynomial which is constructed via a bracket polynomial for tied links. These invariants are more powerful than both the Kauffman and the bracket polynomials…
The topology of -representation varieties of the fundamental groups of planar webs so that the meridians are sent to matrices with trace equal to are explored, and compared to data coming from spider evaluation of the webs. Corresponding to an evaluation of a web as a spider is a rooted tree. We associate t…
Study on the growth of colored Jones polynomial for figure-eight knot cables.
problem Asymptotic behavior of colored Jones polynomial for figure-eight knot cables.
method Analyzing the asymptotic growth of the -dimensional colored Jones polynomial of a cable of the figure-eight knot.
result The growth rate of the colored Jones polynomial is exponential and related to the Chern-Simons invariant.
PER-ETD improves ETD by reducing variance to polynomial complexity.
problem Large variance in ETD leading to exponential sample complexity.
method Periodically restart and update the follow-on trace for a finite period.
result PER-ETD converges to the same fixed point as ETD but with improved sample complexity.
We express the colored Jones polynomial as the inverse of the quantum determinant of a matrix with entries in the -Weyl algebra of -operators, evaluated at the trivial function (plus simple substitutions). The Kashaev invariant is proved to be equal to another special evaluation of the determinant. We also discus…
For a ribbon graph we consider an alternating link in the 3-manifold represented as the product of the oriented surface and the unit interval . We show that the Kauffman bracket is an evaluation of the recently introduced Bollobas-Riordan polynomial . This results generalizes t…
Jones polynomials for knots and links with many crossings calculated efficiently.
problem Computing Jones polynomials for knots and links with a large number of crossings.
method Calculating Tutte polynomials for associated graphs and evaluating with specific substitutions.
result Jones polynomials for knots and links with many crossings calculated efficiently.
Paper extends Cohen's method to compute Jones polynomial for certain braid subfamilies.
problem Computing Jones polynomial for specific knot families.
method Using weighted adjacency matrices and determinants for certain subfamilies of braid groups.
result Jones polynomial can be computed in polynomial time for certain subfamilies of braid groups.
Using a modified foam evaluation, we give a categorification of the Alexander polynomial of a knot. We also give a purely algebraic version of this knot homology which makes it appear as the infinite page of a spectral sequence starting at the reduced triply graded link homology of Khovanov--Rozansky.
New method computes affine normal directions efficiently for sparse polynomials.
problem Computing affine normal directions is computationally expensive in high dimensions.
method Reduces third-order tensor contraction to matrix-free formulation using log-determinant gradient.
result Scalable implementations with near-linear scaling in dimension and sparsity.
Categorifies colored Jones polynomial at roots of unity.
problem Categorification of colored Jones polynomial.
method Differential on triply-graded homology, compatible with p-differential structure.
result Categorification of the colored Jones polynomial at a root of unity.
A new formula approximates knot volume using Jones polynomial evaluations.
problem Approximating the hyperbolic volume of knots using a simple formula.
method Reversing a neural network trained on Jones polynomial evaluations.
result Average error of 2.86% on first 1.7 million knots.
For a hyperbolic knot and a natural number n, we consider the Alexander polynomial twisted by the n-th symmetric power of a lift of the holonomy. We establish the asymptotic behavior of these twisted Alexander polynomials evaluated at unit complex numbers, yielding the volume of the knot exterior. More generally, we pr…
We show how the signed evaluations of link polynomials can be used to calculate unknotting numbers. We use the Jones-Rong value of the Brandt-Lickorish-Millett-Ho polynomial Q to calculate the unknotting numbers of 8_{16}, 9_{49} and 6 further new entries in Kawauchi's tables. Another method is developed by applying an…
We introduce a new cohomology theory for planar trivalent graphs with perfect matchings. The graded Euler characteristic of the cohomology is a one variable polynomial called the 2-factor polynomial that, if nonzero when evaluated at one, implies that the perfect matching is even and therefore the graph is 4-face color…
The article calculates a multiplying factor to convert rational Vassiliev invariants to integer-valued ones.
problem Converting rational valued Vassiliev invariants to integer-valued ones.
method Calculates the minimal multiplying factor λ needed for rational Vassiliev invariants to become integer-valued.
result Obtains a set of integer-valued Vassiliev invariants.
Study shows exponential growth of knot polynomial tied to Chern-Simons invariant.
problem Asymptotic behavior of colored Jones polynomials of figure-eight knot.
method Analyzes growth rate of polynomial evaluated at specific points.
result Growth rate determined by Chern-Simons invariant of an affine representation.
The paper analyzes the asymptotic behavior of a knot polynomial for a specific real number.
problem Understanding the asymptotic behavior of a knot polynomial for a real number.
method Examining the asymptotic behavior of the -dimensional colored Jones polynomial evaluated at for a real number .
result From the asymptotic behavior, the Chern--Simons invariant and the Reidemeister torsion twisted by the adjoint action can be extracted.
New link homologies categorify Jones polynomial at odd prime powers.
problem Categorify Jones polynomial at odd prime powers.
method Specialize Cautis differential to positive integers.
result Non-isomorphic link homologies for odd primes.