New formula simplifies interior polynomial calculation.
problem Calculating interior polynomial efficiently.
method New recursion formula based on non-expanding sets.
result Clearer combinatorial interpretation of interior polynomial.
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48 results for “Polynomial formulas”
Formula found for a specific knot's A-polynomial.
problem Computing the A-polynomial of a specific knot.
method Explicit formula derived for the knot with Conway's notation C(2n, 4).
result The A-polynomial contains exactly the same irreducible factors as the one defined in~\cite{CCGLS1}.
New formulas derived for Jones polynomial of rational links.
problem Calculating the Jones polynomial of rational links.
method Colored Brylawski's tensor product formula for Tutte polynomials, finite automaton for crossing signs.
result Generalization of existing formulas for rational links.
Formulae for Vassiliev invariants derived from Kauffman polynomial.
problem Computing Vassiliev invariants from knot polynomials.
method State model of Kauffman polynomial, Gauss diagram identities, arrow diagram identities.
result Gauss diagram formulae for Vassiliev invariants of order 3.
Formula for HOMFLY polynomial in link diagrams.
problem Calculating HOMFLY polynomial for link diagrams.
method Introduced a class of link diagrams, including closed braids, and proved a generalized full-twist formula.
result Generalized formula for HOMFLY polynomial in new class of link diagrams.
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.
Explicit formulas for pretzel knots' Alexander polynomials.
problem Alexander polynomial of pretzel knots
method Provided explicit formulas
result Characterization of pretzel knots with trivial Alexander polynomial
New formula recovers degree of colored Jones polynomials for pretzel knots.
problem Determining the degree of colored Jones polynomials for specific knots.
method Alternate expansion of the colored Jones polynomial for pretzel links, focusing on 3-tangle knots.
result Determined the degrees of the colored Jones polynomials for a new family of 3-tangle pretzel knots.
Computed formulas for curvature operators and Poincaré polynomials of symmetric spaces.
problem Calculating curvature operators and Poincaré polynomials for symmetric spaces.
method Explicit formulas derived using quantum numbers and eigenvalue analysis.
result Maximum eigenvalue of curvature operators bounded by Einstein constant, with equality for Hermitian spaces.
Formula for computing cross-moments of polynomial processes.
problem Computing cross-moments of polynomial jump-diffusion dynamics.
method Explicit formula based on linear combinations of exponentials of the generator matrix.
result Closed and compact formulations for correlators, useful in financial pricing.
The 2-loop polynomial is a polynomial presenting the 2-loop part of the Kontsevich invariant of knots. We show a cabling formula for the 2-loop polynomial of knots. In particular, we calculate the 2-loop polynomial for torus knots.
We derive formulas for HOMFLY polynomials of torus links using braid groups and linear recurrences.
problem Calculating HOMFLY polynomials for torus links.
method Using braid groups and linear recurrences, derived from the skein relation.
result Explicit formulas for HOMFLY polynomials of torus links and are derived.
Formula for colored Links-Gould polynomial with genus bounds.
problem Calculating polynomial for knots colored with specific representations.
method Cabling formula and genus bounds for the Links-Gould polynomial.
result Genus bounds and specialization to Alexander polynomial for colored Links-Gould polynomial.
Formula for 2-bridge link Alexander polynomials via integer walks.
problem Computing Alexander polynomials of 2-bridge links.
method Extending Hartley's and Minkus' formula to 2-variable polynomials via integer lattice walks.
result A formula corresponding to a walk on the 2D integer lattice.
New method proves Lickorish-Millett formulae for link polynomials.
problem Proving Lickorish-Millett type formulae for links.
method Introducing a new method to prove the formulae.
result New method successfully proves the formulae.
Proves a formula for Kontsevich-Witten tau-function using Schur Q-polynomials.
problem Proving the Kontsevich-Witten tau-function formula.
method Directly shows Q-polynomial expansion satisfies Virasoro constraints.
result Direct proof of the formula without matrix model.
The paper calculates colored Jones polynomials for 2-bridge links using skein theory.
problem Calculating colored Jones polynomials for 2-bridge links.
method Graphical calculus and skein theory.
result Explicit calculation of and colored Jones polynomials for 2-bridge links.
Rosso and Jones gave a formula for the colored Jones polynomial of a torus knot, colored by an irreducible representation of a simple Lie algebra. The Rosso-Jones formula involves a plethysm function, unknown in general. We provide an explicit formula for the second plethysm of an arbitrary representation of $\fsl_3$, …
Study Alexander polynomials of ribbon and virtual knots using ribbon's intrinsic singularity.
problem Determining Alexander polynomials for ribbon and virtual knots.
method Using ribbon's intrinsic singularity information, defining half Alexander polynomial, and developing simplified formulas.
result New formulas for Alexander polynomials of general knots and virtual knots in terms of Gauss diagrams.
An explicit formula for the -polynomial of the knot with Conway's notation is obtained from the explicit Riley-Mednykh polynomial of it.
In this paper, a generalized version of Morton's formula is proved. Using this formula, one can write down the colored Jones polynomials of cabling of an knot in terms of the colored Jones polynomials of the original knot.
Using Chebyshev polynomials, C. Frohman and R. Gelca introduce a basis of the Kauffman bracket skein module of the torus. This basis is especially useful because the Jones-Kauffman product can be described via a very simple Product-to-Sum formula. Presented in this work is a diagrammatic proof of this formula, which em…
We extend Hoste-Shanahan's calculations for the A-polynomial of twist knots, to give an explicit formula.
The paper studies twisted Alexander polynomials for knot groups in various extensions.
problem Understanding twisted Alexander polynomials in knot groups for different extensions.
method Developed mod p formula for twisted Alexander polynomials and studied central extensions.
result Established formulas for twisted Alexander polynomials in knot groups for various extensions.
Formula for interior polynomial of bipartite graphs derived from knot theory.
problem Deriving a formula for the interior polynomial of bipartite graphs.
method Applied knot theory, Ehrhart reciprocity, flyping and mutation.
result Proved a mirroring formula for the interior polynomial of bipartite graphs.
Paper computes Alexander polynomials for arborescent links.
problem Explicit formulas for Alexander polynomials are hard to compute for most link families.
method Efficient method for arborescent links, using recursive polynomials.
result Explicit closed formulas for pretzel links derived.
In analogy with a recursive formula for the HOMFLY-PT polynomial of links given by Jaeger, we give a recursive formula for the graph polynomial introduced by Kauffman and Vogel. We show how this formula extends to the Khovanov-Rozansky graph homology.
Character variety of Borromean link solved, Alexander polynomial formula found.
problem Character variety of Borromean link
method Determined irreducible SL(2,C) character variety, found Alexander polynomial formula
result Formula for twisted Alexander polynomial on character variety
Formula for twisted Alexander polynomials of torus links.
problem Computing twisted Alexander polynomials for torus links.
method Explicit formula derivation and character variety analysis.
result Locally constant function on character variety.
We compute q-holonomic formulas for the HOMFLY polynomials of 2-bridge links colored with one-column (or one-row) Young diagrams.
Two formulas for Chern classes of tensor products of vector bundles are presented.
problem Calculating Chern classes for tensor products of vector bundles.
method Two formulas using matrices and polynomials to compute Chern classes.
result Determinantal formulas for Chern classes of tensor products.
We describe the Polyak-Viro arrow diagram formulas for the coefficients of the Conway polynomial. As a consequence, we obtain the Conway polynomial as a state sum over some subsets of the crossings of the knot diagram. It turns out to be a simplification of a special case of Jaeger's state model for the HOMFLY polynomi…
This paper studies HOMFLY polynomials of specific and infinite classes of knots.
problem Computing HOMFLY polynomials in general is difficult; this paper examines specific cases.
method Examined two specific knots and a general infinite class of knots.
result Observed apparent patterns in the polynomials of specific knots and conjectured properties of the general class.
Paper proves a generalized Torres formula for twisted Reidemeister torsion.
problem Alexander polynomial properties for links and sublinks.
method Uses twisted Reidemeister torsion to generalize Torres formula.
result Obtains a second proof of Morifuji's result.
Formula for 2-head of colored Jones polynomial for pretzel knots proved.
problem Calculating the 2-head of colored Jones polynomial for pretzel knots.
method Skein-theoretic techniques and stability properties of coefficients.
result Formula for 2-head of colored Jones polynomial proved for pretzel knots.
Poincare series of Klein groups, Coxeter polynomials, the Burau representation and Milnor invariantsmath.GT
For Poincare series of binary polyhedral groups and Coxeter polynomials there are obtained statements close to the Euclid algorithm and orthogonal polynomials theory: generalized Ebeling formula, decompositions into ramified continued fractions, Christoffel-Darboux identity, combinatorial formula. Known results about t…
The colored HOMFLY polynomial is the quantum invariant of oriented links in associated with irreducible representations of the quantum group . In this paper, using an approach to calculate quantum invariants of links via cabling-projection rule, we derive a formula for the colored HOMFLY polyn…
The paper connects curvature data to polynomial coefficients in gluing formulas.
problem Understanding polynomial coefficients in gluing formulas for zeta-determinants.
method Expressing coefficients of a polynomial in terms of scalar and principal curvatures of a 2D hypersurface.
result Coefficients of the polynomial are expressed in terms of curvature data.
Diagrammatic calculus proves Alexander polynomial formulas.
problem Proving formulas for Alexander polynomial.
method Diagrammatic calculus of quantum representations.
result Quantum invariant determines Alexander polynomial.
Paper describes a state sum formula for a graph coloring polynomial.
problem Counting -face colorings of ribbon graphs for various .
method Combines topological quantum field theory and diagrammatic tensors.
result Describes a state sum formula for the total face color polynomial.
We develop a diagrammatic formalism for calculating the Alexander polynomial of the closure of a braid as a state-sum. Our main tools are the Markov trace formulas for the HOMFLY-PT polynomial and Young's semi-normal representations of the Iwahori-Hecke algebras of type A.
We explore Jaeger's state model for the HOMFLYPT polynomial. We reformulate this model in the language of Gauss diagrams and use it to obtain Gauss diagram formulas for a two-parameter family of Vassiliev invariants coming from the HOMFLYPT polynomial. These formulas are new already for invariants of degree 3.
The taut polynomial equals a twisted Alexander polynomial.
problem Understanding the relationship between taut polynomials and Alexander polynomials.
method Defined taut polynomial of veering triangulations and proved it equals a twisted Alexander polynomial.
result The taut polynomial equals a twisted Alexander polynomial of the underlying manifold.
New formula for 3-manifold invariants using combinatorial methods.
problem Calculating invariants of 3-manifolds.
method Diagrammatic and combinatorial approach, focusing on the Conway polynomial and Kontsevich integral.
result A general surgery formula for the Casson-Walker-Lescop invariant.
Formulae for Yamada polynomial of spatial graphs are derived from edge replacements.
problem Computing Yamada polynomial for spatial graphs formed by edge replacements.
method Formulae derived from edge replacements of plane graphs.
result Zeros of Yamada polynomials of certain spatial graphs are dense in a complex plane region.
Formula for Alexander polynomial of links with twists.
problem Computing Alexander polynomial of links with twists.
method Using vector space representation of .
result Alexander polynomials stabilize after adding enough twists.
We derive formulas for Alexander polynomials of spiral knots.
problem Understanding spiral knots, a braid-theoretic generalization of torus knots.
method Recursive formula for Alexander polynomials, genus formula.
result Simple genus formula for spiral knots.
We conjecture formulae of the colored superpolynomials for a class of twist knots where p denotes the number of full twists. The validity of the formulae is checked by applying differentials and taking special limits. Using the formulae, we compute both the classical and quantum super-A-polynomial for the twist k…