The paper introduces new structures for colored HOMFLY-PT invariants using skein theory.
problem Proving strong integrality and deriving symmetric properties for HOMFLY-PT invariants.
method Purely using HOMFLY-PT skein theory and applying to LMOV conjecture.
result Strong integrality and symmetric properties for colored HOMFLY-PT invariants.
Colored HOMFLY-PT invariant, the generalization of the colored Jones polynomial, is one of the most important quantum invariants of links. This paper is devoted to investigating the basic structures of the colored HOMFLY-PT invariants of links. By using the HOMFLY-PT skein theory, firstly, we show that the (reformulate…
Conjectures closed-form expressions and cyclotomic expansions for knot invariants.
problem Calculating HOMFLY-PT invariants of knots colored by rectangular diagrams.
method Interpolation Macdonald polynomials and cyclotomic expansions.
result Conjectured closed-form expressions and cyclotomic expansions for knot invariants.
Proved colored HOMFLY-PT polynomials for specific knots.
problem Calculating colored HOMFLY-PT polynomials for specific knots.
method Rigorous mathematical proof for trefoil, figure-eight, and twist knots.
result Colored HOMFLY-PT polynomials expressed as sums for different knots.
We define reduced colored sl(N) link homologies and use deformation spectral sequences to characterize their dependence on color and rank. We then define reduced colored HOMFLY-PT homologies and prove that they arise as large N limits of sl(N) homologies. Together, these results allow proofs of many aspects of the phys…
Proved cyclotomic expansion for double twist knots' HOMFLY-PT invariants.
problem Proving cyclotomic expansion for colored HOMFLY-PT invariants of double twist knots.
method Cyclotomic expansion formula for double twist knots.
result Confirmed cyclotomic expansion conjecture for SU(N)-invariants.
New method for calculating colored HOMFLY-PT polynomials for links with different symmetric representations.
problem Calculating colored HOMFLY-PT polynomials for links with arbitrary symmetric representations.
method Using quantum Racah coefficients (6j-symbols) of Uq(sl2) to simplify the evaluation. result Multi-colored link polynomials H[r1],[r2] for a specific link L7a3 are successfully evaluated. 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.
New symmetry found in colored Alexander polynomial.
problem Understanding the structure of colored Alexander polynomials.
method Study of loop and character expansions, group theoretic constraints.
result Existence of a new symmetry in the colored HOMFLY-PT polynomial.
We connect knot contact homology to colored HOMFLY-PT polynomials using SFT and recursion.
problem Understanding colored HOMFLY-PT polynomials for knots and links.
method Legendrian Symplectic Field Theory, large N duality, Witten's connection, induction, elimination theory. result Established a recursion relation for colored HOMFLY-PT polynomials using SFT and elimination theory.
New categorification method for infinite braids.
problem Categorifying highest-weight projectors for infinite braids.
method Limiting colored Khovanov-Rozansky homology of infinite braids.
result Partial isomorphism between HOMFLY-PT homology of braids and infinite torus knots.
Topological recursion recovers a specific partition function for colored knots.
problem Recovering the extended Ooguri-Vafa partition function for colored HOMFLY-PT polynomials of torus knots.
method Applying topological recursion to the spectral curve of colored HOMFLY-PT polynomials of torus knots.
result Topological recursion reproduces the n-point functions of the extended Ooguri-Vafa partition function.
We study various specializations of the colored HOMFLY-PT polynomial. These specializations are used to show that the multivariable link invariants arising from a complex family of sl(m|n) super-modules previously defined by the authors contains both the multivariable Alexander polynomial and Kashaev's invariants. We c…
Aicardi's invariant F(L) is extended to colored singular links using graphical calculus.
problem Constructing an invariant for colored classical and singular links.
method State-sum model using graphical calculus for oriented, colored, 4-valent planar graphs.
result Extends F(L) to colored singular links, showing it's stronger than HOMFLY-PT polynomial. This work reconstructs knot invariants from Alexander polynomials, proving consistency with known theorems.
problem Reconstructing knot invariants from Alexander polynomials.
method Quantization, deformation, and rewriting of Alexander polynomials.
result Derives new formulae for colored superpolynomials and proves consistency with Melvin-Morton-Rozansky theorem.
New formulas link knot invariants from DGA and satellite polynomials.
problem Establishing relationships between knot invariants.
method Introducing new polynomials and formulas linking DGA and satellite invariants.
result Arbitrary m-graded satellite ruling polynomials are determined by DGA of K.
The abstract conjectures a link between knot homologies and quiver partition functions.
problem Understanding the relationship between knot homologies and quiver partition functions.
method Interpreting quiver nodes as holomorphic curves with boundary on the knot conormal, and studying recursion relations.
result Generalized quiver partition functions are related to knot homologies.
Study on distinguishing mutant knots using specific representations.
problem Distinguishing mutant knots using colored HOMFLY-PT polynomials.
method Calculating polynomials and differences for mutant knot polynomials in specific representations.
result Properties of mutant knot polynomials in representations [3,1] and [4,2] were studied.
The paper connects knot homology, quantum 6j-symbols, and complements of knots.
problem Investigating the relationship between knot homology, quantum 6j-symbols, and knot complements.
method Developed a grading rule for HOMFLY-PT and Kauffman homology, found relationships between A-polynomials, and conjectured closed-form expressions for quantum 6j-symbols and knot complements.
result Closed-form expressions for SO(N) quantum 6j-symbols and conjectured expressions for (a,t)-deformed F_K for knot complements.
Researchers map knot complements using 3d theories and half-index calculations.
problem Mapping knot complements using mathematical theories.
method Using 3d N=2 theories and half-index calculations. result Realized homological blocks and HOMFLY-PT polynomials for knot complements.
Study lattice paths from twist knots and double twist knots.
problem Understanding combinatorics of twist knots and double twist knots.
method Analyzing quiver generating series of HOMFLY-PT polynomial limits.
result Lattice path models for twist knots and double twist knots.
We first study superpolynomial associated to triply-graded reduced colored HOMFLY-PT homology. We propose conjectures of congruent relations and cyclotomic expansion for it. We prove conjecture of N=1 for torus knot case, through which we obtain the corresponding invariant α(T(m,n))=−(m−1)(n−1)/2. This is closely r…
Defect of knot polynomials remains invariant under certain braid substitutions.
problem Invariance of knot polynomial defects under specific transformations.
method Investigation of defect invariants under antiparallel and parallel braid substitutions.
result Defect remains unchanged under antiparallel braid substitutions and changes by half the added length under parallel braid substitutions.
New geometric proof for rational tangles links-quivers correspondence.
problem Recovering symmetric/antisymmetric colored HOMFLY-PT polynomials from a quiver.
method Geometric approach using winding numbers in punctured plane and its second configuration space.
result Explicit description of quivers for rational tangles.
The usual construction of link invariants from quantum groups applied to the superalgebra D_{2 1,alpha} is shown to be trivial. One can modify this construction to get a two variable invariant. Unusually, this invariant is additive with respect to connected sum or disjoint union. This invariant contains an infinity of …
Geometrically describes the linear and quadratic forms for rational links.
problem Predicting generating functions for colored HOMFLY-PT polynomials of rational links.
method Direct geometric description of linear and quadratic forms in terms of configuration spaces.
result Direct geometric description of forms for rational links.
Study proves Hecke lifting conjecture for torus knots and verifies it for any framed knots.
problem Integrality structure of framed knots' quantum invariants.
method Explicit formulas of colored HOMFLY-PT invariants of torus knots, verified in a limit form for any framed knots.
result Proves Hecke lifting conjecture for torus knots and verifies it for any framed knots.
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 q∣R∣. M. Khovanov and L. Rozansky gave a categorification of the HOMFLY-PT polynomial. This study is a generalization of the Khovanov-Rozansky homology. We define a homology associated to the quantum (sln,∧Vn) link invariant, where ∧Vn is the set of the fundamental representations of the quantum group of $sl…
We use super q-Howe duality to provide diagrammatic presentations of an idempotented form of the Hecke algebra and of categories of glN-modules (and, more generally, glN∣M-modules) whose objects are tensor generated by exterior and symmetric powers of the vector representations. As an ap…
Legendrian knot representations linked to colored Kauffman polynomial.
problem Relating Legendrian knot representations to colored Kauffman polynomial.
method Introducing ungraded n-colored ruling polynomial and relating it to the n-colored Kauffman polynomial. result Ungraded representation numbers of Legendrian knot DG-algebra agree with n-colored Kauffman polynomial specialization. New insights into non-torus links using topological vertices.
problem Understanding non-torus links through topological vertices.
method Tangle calculus and topological string considerations.
result Explicit description of a non-torus link L8n8. We introduce a new class of quantum enhancements we call biquandle brackets, which are customized skein invariants for biquandle colored links.Quantum enhancements of biquandle counting invariants form a class of knot and link invariants that includes biquandle cocycle invariants and skein invariants such as the HOMFLY…
Genus 2 mutation is the process of cutting a 3-manifold along an embedded closed genus 2 surface, twisting by the hyper-elliptic involution, and gluing back. This paper compares genus 2 mutation with the better-known Conway mutation in the context of knots in the 3-sphere. Despite the fact that any Conway mutation can …
Computes knot types using HOMFLY-PT polynomial.
problem Determining chiral knot and link types with small crossing numbers.
method Uses the HOMFLY-PT polynomial to compute knot types from 3D coordinates.
result Efficacy of HOMFLY-PT for knot types up to crossing number 16.
The paper calculates R and Racah matrices for SO(5) and finds Kauffman polynomials.
problem Generalizing Reshetikhin-Turaev approach to SO(2n+1) case.
method Provided R and Racah matrices for SO(5) symmetric representation.
result Found Kauffman polynomials for SO(5) symmetric representation.
Dihedral linking invariant uses knot colorings to distinguish knots.
problem Distinguishing knots using knot colorings and linking numbers.
method Algorithm for computing linking numbers in dihedral branched covers.
result The dihedral linking invariant distinguishes more than 98% of prime knot pairs.
New algorithm for HOMFLY-PT polynomial reduces computation time.
problem Computing HOMFLY-PT polynomial is #P-hard.
method Fixed-parameter tractability in treewidth.
result HOMFLY-PT polynomial can be computed efficiently using sub-exponential time algorithm.
Khovanov and Rozansky's categorification of the HOMFLY-PT polynomial is invariant under braidlike isotopies for any link diagram and Markov moves for braid closures. To define HOMFLY-PT homology, they required a link to be presented as a braid closure, because they did not prove invariance under the other oriented Reid…
Extends knot invariant computation to symmetrically colored sl_N.
problem Computing quantum knot invariants for slN. method Develops symmetrically colored R matrix for slN. result Defines FKslN,sym for positive braid knots. Study HOMFLY-PT homology structure for knots up to 11 crossings.
problem Understanding the structure of HOMFLY-PT homology for knots.
method Using Nakagane and Sano's knot data and the sl(2) action, compute HOMFLY-PT S-invariant and compare to sl(N) invariants. result Computed HOMFLY-PT S-invariant for all knots in the dataset. Topological model created for HOMFLY-PT polynomial from link diagrams.
problem Constructing categorifications for HOMFLY-PT polynomial.
method Explicit Lagrangian submanifolds on Heegaard surfaces.
result Invariants derived from link diagrams are given by intersections of submanifolds.
Formula for Dehn twists on HOMFLY-PT skein modules with applications.
problem Understanding the action of Dehn twists on HOMFLY-PT skein modules.
method Introduced a formula for the action of Dehn twists on the HOMFLY-PT type skein module of a surface.
result Constructed an invariant \( z(M) \) for integral homology 3-spheres, finite type invariant of order \( n \).
In \cite{GZ}, Gilmer and Zhong established the existence of an invariant for links in S1×S2 which is a rational function in variables a and s and satisfies the HOMFLY-PT skein relations. We give formulas for evaluating this invariant in terms of a standard, geometrically simple basis for the HOMFLY-PT ske…
Constructs y-ifications of Khovanov homology and proves compatibility with HOMFLY--PT.
problem Distinguishing knots with identical Khovanov and HOMFLY--PT homologies.
method Elementary construction within Bar-Natan's framework for tangles, defining e-action on y-ifications. result New structures distinguish knots with identical homologies, e.g., Conway and Kinoshita-Terasaka knots.
A new method calculates HOMFLY-PT polynomials for bipartite links.
problem Computing HOMFLY-PT polynomials for bipartite links efficiently.
method Generalizes Goeritz matrix method for bipartite links.
result Reduces HOMFLY-PT polynomial calculation to matrix algebra.
Computes Khovanov homology for 2-strand braids via graph relations.
problem Computing Khovanov homology for complex links.
method Combinatorial approach using trivalent plane graphs and relations.
result Computation of Khovanov-Rozansky homology for 2-strand braid links.
Knot diagrams can have isomorphic homologies, challenging HOMFLY-PT theory.
problem Non-determinacy of HOMFLY-PT homology for knot diagrams.
method Expands on Abel's method [Abe17].
result Isomorphic homologies for non-deterministic knot diagrams.