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.
Paper develops new methods linking quantum mechanics and finance.
problem Modeling financial markets using quantum mechanics.
method Inverts quantum stochastic processes to nonlocal diffusions, uses path integral formalism and PT symmetry.
result Non-Gaussian kernel function for Accardi-Boukas quantum Black-Scholes.
We define composite DAHA-superpolynomials of torus knots, depending on pairs of Young diagrams and generalizing the composite HOMFLY-PT polynomials in the theory of the skein of the annulus. We provide various examples. Our superpolynomials extend the DAHA-Jones (refined) polynomials and satisfy all standard symmetries…
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…
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 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.
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…
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.
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.
PT-WGAN reduces PET image noise without losing details.
problem Low-dose PET images suffer from noise and artifacts.
method Parameter-transferred Wasserstein GAN for denoising.
result PT-WGAN outperforms state-of-the-art methods in denoising.
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.
This work challenges the assumption that shorter conformal prediction intervals are always better.
problem The conventional evaluation of conformal prediction metrics (coverage and interval length) may not fully capture the quality of predictions.
method The Prejudicial Trick (PT) is introduced, which probabilistically returns either a null interval or a longer one to maintain valid coverage while potentially reducing interval length.
result The Prejudicial Trick can yield deceptively shorter intervals without compromising coverage, but introduces practical vulnerabilities.
Researchers establish a connection between knot homology and Lie algebra actions.
problem Understanding the HOMFLY-PT homology of (n,n+1) torus knots. method Constructing an explicit isomorphism and computing tautological class actions.
result The tautological class action extends to Hamiltonian vector fields and differentials in spectral sequences.
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.
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.
We examine the relationship between the (untwisted) knot Floer cube of resolutions and HOMFLY-PT homology. By using a filtration induced by additional basepoints on the Heegaard diagram for a knot K, we see that the filtered complex decomposes as a direct sum of HOMFLY-PT homologies of various subdiagrams. Jaeger's c…
2D-PT improves sampling in constrained optimization problems.
problem Sampling Boltzmann distributions with soft constraints.
method Two-dimensional extension of parallel tempering.
result 2D-PT achieves near-ideal mixing in constrained problems.
In this paper we compute the reduced HOMFLY-PT homologies of the Conway and the Kinoshita-Terasaka knots and show that they are isomorphic.
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.
We show that for any Legendrian link L in the 1-jet space of S1 the 2-graded ruling polynomial, RL2(z), is determined by the Thurston-Bennequin number and the HOMFLY-PT polynomial. Specifically, we recover RL2(z) as a coefficient of a particular specialization of the HOMFLY-PT polynomial. Furthermore, …
HZ transform applied to knot polynomials reveals hyperbolic knot structures.
problem Understanding the structure of knot polynomials and their factorisability.
method Applying the Harer-Zagier transform to knot polynomials and character expansions.
result Construction of an infinite family of hyperbolic knots and proof of factorisability in the 3-strand case.
Machine learning assesses balance outside clinics, improving therapy efficiency.
problem Lack of feedback from PTs in home balance training.
method Trunk sway data analysis with multi-class SVM.
result ML model achieved 82% accuracy in assessing balance.
New findings on Jones polynomial for 4-strand braids.
problem Whether there are non-trivial knots with trivial Jones polynomial.
method Study of 4-strand braids, exploration of various properties of hypothetical HOMFLY-PT polynomials.
result Existence of a 1-parameter family of 2-variable polynomials that can be HOMFLY-PT polynomials of some 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…
Using the diagrammatic calculus for Soergel bimodules, developed by B. Elias and M. Khovanov, as well as Rasmussen's spectral sequence, we construct an integral version of HOMFLY-PT and sl(n)-link homology.
Proposes PT-MMD for evaluating generative models.
problem Evaluating generative models under implementation constraints.
method Combines MMD and PT resampling for statistical evaluation.
result Demonstrates effectiveness in model selection and image fidelity.
New method accelerates Parallel Tempering using neural samplers.
problem Challenges in sampling from high-dimensional, multimodal distributions.
method Leverages neural samplers to reduce overlap between distributions.
result Improves sample quality and reduces computational cost.
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.
Let EkF(D) be the spectral sequence induced by the oriented cube of resolutions on knot Floer homology. We prove that E2F(D) is a triply graded link invariant whose graded Euler characteristic is the HOMFLY-PT polynomial and that the higher pages are link invariants. By construction, the spectral sequen…
PT network optimizes asset weights without forecasting returns.
problem Traditional asset allocation methods are error-prone and limit portfolio performance.
method PT network uses attention mechanisms to directly optimize Sharpe ratio.
result PT outperforms other algorithms in risk-adjusted performance.
Study improves dynamic PT fleet optimization under noisy demand predictions.
problem Accurately predicting dynamic public transport demand for effective fleet management.
method Experimental case study in Copenhagen, using linear programming to optimize fleets.
result Optimized fleet performance is mainly affected by noise distribution skew and large errors.
A strategy for spectrum sharing in CRNs with multiple PT power levels.
problem Efficient spectrum usage for secondary users in CRNs with multiple PT power levels.
method Data-driven/machine learning based multi-level spectrum sensing and prediction-transmission structures.
result The proposed strategy effectively aligns the ST with the PT power levels, improving spectrum usage.
The (untwisted) oriented cube of resolutions for knot Floer homology assigns a complex CF(S) to a singular resolution S of a knot K. Manolescu conjectured that when S is in braid position, the homology H∗(CF(S)) is isomorphic to the HOMFLY-PT homology of S. Together with a naturality condition on t…
Let (N,g) be an n-dimensional complete Riemannian manifold with nonempty boundary $\pt N$. Assume that the Ricci curvature of N has a negative lower bound Ric≥−(n−1)c2 for some c>0, and the mean curvature of the boundary $\pt N$ satisfies H≥(n−1)c0>(n−1)c for some c0>c>0. Then a known result (s…
New algebraic setup defines quantum link invariants.
problem Defining and controlling quantum link invariants.
method Quantum Schur--Weyl duality and variants.
result Global definitions of quantum polynomials.
For a ring R, we denote by R[L] the free R-module spanned by the isotopy classes of singular links in S3. Given two invertible elements x,t∈R, the HOMFLY-PT skein module of singular links in S3 (relative to the triple (R,t,x)) is the quotient of R[L] by local rela…
We connect Causal inference and low-rank recovery via RDT and free probability theory.
problem Determining the applicability of causal inference via low-rank recovery.
method Random Duality Theory, free probability theory, and mathematical rigor.
result Exact closed-form worst case phase transitions for causal inference.
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.
DEPTS learns to forecast periodic time series with improved accuracy.
problem Forecasting periodic time series is challenging due to complex dependencies and diverse periods.
method DEPTS uses a decoupled formulation with an expansion module and a periodicity module to handle these challenges.
result DEPTS significantly improves forecasting accuracy, reducing errors by up to 20%.