New method computes triply graded link homology for torus links.
problem Computing triply graded link homology for torus links.
method Introducing a new method for triply graded link homology specifically adapted to torus links.
result Exact answers for (n,n)-torus links, verifying conjectures about homology and Hilbert schemes.
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.
Researchers compute link surgery modules for 2-component L-space links.
problem Computing the link surgery modules of 2-component L-space links.
method Koszul duality to extend earlier computations.
result Computed entire link surgery modules modulo a technical result.
Paper computes link determinants using Fourier-Hadamard transforms.
problem Computing determinants of complex link structures.
method Fourier-Hadamard transforms of Boolean functions.
result Determinant of centrally symmetric links with even components equals zero.
New method uses algebras to speed up link Floer homology calculations.
problem Computing link Floer homology efficiently.
method Using bordered algebras to compute link Floer homology.
result Fast computation of the Thuston polytope for links.
Simple algorithm for computing link polynomials using skein relations.
problem Computing link polynomials efficiently and accurately.
method New total order on braid representatives to define and compute link polynomials.
result New complete link invariant as a by-product of the algorithm.
Computed linking number of modular knots and Lorenz links.
problem Computing the linking number of modular knots in a specific space.
method Using correspondence between modular links and Lorenz links, and intersection number involving Conway topographs.
result Computed linking number of modular knots and compared to a previous formula.
Study homology of torus links and colored knots.
problem Computing homology of specific link types.
method Khovanov-Rozansky homology for positive torus links and colored knots.
result Computed homology for a family of links including torus links and colored knots.
Study on finite n-quandles of knots and links, computing graphs and groups.
problem Understanding finite n-quandles of specific knot types.
method Computed Cayley graphs and automorphism groups.
result Detailed data for two-bridge and torus knots and links.
Computed A-polynomial of twisted Whitehead links.
problem Calculating the A-polynomial for a specific class of links.
method Direct computation of the A-polynomial using mathematical techniques.
result Determined canonical components and volume formula for twisted Whitehead links.
New invariant for links in 3-sphere computed and computed using diagrams.
problem Computing invariants of links in 3-sphere.
method Defining and computing the parabolic Dijkgraaf-Witten invariant.
result Computed invariants of several links and partial information using diagrams.
Formula for computing rotation and self-linking numbers in contact surgery diagrams.
problem Computing rotation and self-linking numbers for Legendrian knots and transverse knots in contact surgery diagrams.
method Explicit formula and extension of Ding-Geiges-Stipsicz formula for d3-invariant.
result Explicit formulas for rotation and self-linking numbers in contact (1/n)-surgery diagrams.
Computed SU(2) Casson-Lin invariant for Hopf link.
problem Computing the SU(2) Casson-Lin invariant for specific links. method Direct computation and analysis of the Hopf link.
result Determined the sign in the linking number formula.
Efficient method for computing twisted Alexander polynomials of Montesinos links.
problem Computing twisted Alexander polynomials for Montesinos links efficiently.
method Developed an efficient method to compute the twisted Alexander polynomial for Montesinos links using any linear representation.
result Formulas for multi-variable Alexander polynomials can be easily derived.
We develop the intersection theory at relative chain-cochain level, and apply it along with the use of Seifert disks for an oriented link to give a combinatorial algorithm to compute Massey's higher order linking numbers. It is subtle to compute higher-order linking numbers, and it has been a folklore to use the inters…
Study signatures of torus links and their cores using Neumann's equivariant signatures and Hirzebruch's formula.
problem Computing signatures of torus links and their cores.
method Use Neumann's equivariant signatures and rewrite Hirzebruch's formula for torus links (without cores) in terms of integral points in a parallelogram.
result Rewritten Hirzebruch's formula for torus links with cores using integral points in a parallelogram.
A new method computes link invariants from diagrams.
problem Computing link invariants efficiently.
method Single symmetric matrix from a link diagram.
result Multivariable Alexander polynomial computation.
New method speeds up knot computations in 3D.
problem Computational complexity in knot theory.
method 3D representation of knots for faster computation.
result Savings in computational complexity for knot invariants.
Researchers calculate link Floer homology for 2-component L-space links.
problem Computing link Floer homology for specific L-space links.
method Using the h-function of the filtered chain complex determined by Alexander polynomials. result Explicit determination of Thurston polytope and norm for 2-component L-space links.
The paper tabulates and computes the number of alternating pretzel links up to a given crossing number.
problem Computing the total number of alternating pretzel links for a given crossing number.
method Derived a closed formula to compute the total number of alternating pretzel links, P(c), for any given crossing number c. result The number of alternating pretzel links grows exponentially with the crossing number.
The article computes the Blanchfield pairing for any link.
problem Computing the Blanchfield pairing for arbitrary links.
method Explicit computation for any link.
result Explicit formula for the Blanchfield pairing of any link.
Reconfigures Milnor invariants using unipotent Magnus embeddings.
problem Computing and understanding Milnor invariants of links.
method Central group extensions and unipotent Magnus embeddings.
result First non-vanishing invariants and refinements of higher degree invariants computed.
New class of links with specific homology properties and instanton computations.
problem Understanding homology properties of links and their instanton invariants.
method Introduced a new class of links, computed instanton homology, and discussed spectral sequences.
result Computed framed instanton homology for double branched covers of new links.
Using computer calculations and working with representatives of pretzel tangles we established general adequacy criteria for different classes of knots and links. Based on adequate graphs obtained from all Kauffman states of an alternating link we defined a new numerical invariant: adequacy number, and computed adequac…
Wirtinger number equals virtual bridge number for virtual links.
problem Calculating the virtual bridge number of virtual links.
method Algorithmically computing the minimum number of generators of the link group.
result The Wirtinger number equals the virtual bridge number for virtual links.
We show that nontrivial classical pretzel knots L(p,q,r) are hyperbolic with eight exceptions which are torus knots. We find Conway polynomials of n-pretzel links using a new computation tree. As applications, we compute the genera of n-pretzel links using these polynomials and find the basket number of pretzel links b…
We describe an algorithm for computing boundary slopes of 2-bridge links. As an example, we work out the slopes of the links obtained by 1/k surgery on one component of the Borromean rings. A table of all boundary slopes of all 2-bridge links with 10 or less crossings is also included.
The splitting number of a link is the minimal number of crossing changes between different components required, on any diagram, to convert it to a split link. We introduce new techniques to compute the splitting number, involving covering links and Alexander invariants. As an application, we completely determine the sp…
The paper improves bounds on the complexity of computing link polynomials.
problem Computing link polynomials by the skein relation is complex.
method Proved new upper and lower bounds on skein tree depth.
result New bounds on skein tree depth are stronger than previous ones.
Paper introduces simplified formulas for Milnor's triple linking number.
problem Computational difficulty in calculating Jones polynomial for topological polymers.
method Developed Gauss diagram formulas for Milnor's Vassiliev invariants.
result Introduced non-torsion valued Milnor's triple linking number.
Computing unlinking number is usually very difficult and complex problem, therefore we define BJ-unlinking number and recall Bernhard-Jablan conjecture stating that the classical unknotting/unlinking number is equal to the BJ-unlinking number. We compute BJ-unlinking number for various families of knots and links for w…
Technical report classifies all principal congruence link groups.
problem Classifying all principal congruence link complements in S^3.
method Comprehensive case analysis, necessary computations, and computer programs.
result Complete classification of all principal congruence link groups.
Computed minimum crossing numbers for Turaev genus 2 links.
problem Verifying the Qazaqzeh-Chbili-Lowrance conjecture.
method Computed minimum crossing numbers for a specific family of links.
result Verified the Qazaqzeh-Chbili-Lowrance conjecture for the family.
We present computational results about quasi-alternating knots and links and odd homology obtained by looking at link families in the Conway notation. More precisely, we list quasi-alternating links up to 12 crossings and the first examples of quasi-alternating knots and links with at least two different minimal diagra…
The paper studies Turaev genus one links and finds their Khovanov homology to be isomorphic to Z in at least one extremal grading.
problem Understanding the extremal properties of Turaev genus one links.
method Analyzing Khovanov homology of Turaev genus one links.
result Khovanov homology of Turaev genus one links is isomorphic to Z in at least one extremal grading.
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.
The study explores if a link can be transformed into another using specific diagram manipulations.
problem Can a link be obtained from another using crossing exchanges and smoothings?
method The problem is approached from a computational complexity perspective, focusing on specific link types.
result For certain types of links (torus links and twist knots), there is an algorithm to determine if a link can be transformed into another using crossing exchanges and smoothings in polynomial time.
For families of knots and links given in Conway notation we compute lower maximal and upper minimal bound of hyperbolic volume by using source links and augmented links.
In-degree quiver polynomials for surface-links computed.
problem Computing in-degree quiver polynomials for surface-links.
method Defined using a quandle and set of endomorphisms, computed for surface-links with ch-index up to 10.
result Example computations for surface-links with ch-index up to 10.
The paper computes Alexander and Jones polynomials for surgerized tst links.
problem Computing polynomial invariants for a specific class of links.
method Generalized the notion of tst links and performed specific operations to create new links.
result Computed Alexander and Jones polynomials for surgerized tst links.
Paper computes knot symmetric quandle for surface-links and finds infinitely many distinct surface-knots.
problem Computing knot symmetric quandle for surface-links.
method Using plat form presentations, the paper computes the knot symmetric quandle for surface-links.
result Infinitely many distinct surface-knots of genus g with plat indices m.
We compute the average Tristram---Levine signature of any graph link with positive weights in a three sphere, generalizing the results of Kirby and Melvin. The main tools are the Neumann's algorithm for computing the equivariant signatures of graph links and the Reciprocity Law for Dedekind sums.
Computes Blanchfield pairing for colored links with non-zero Alexander polynomial.
problem Computing Blanchfield pairing for colored links with specific properties.
method Uses generalized Seifert matrices derived from C-complexes.
result Expresses Blanchfield pairing in terms of these matrices.
We use a link invariant defined by Cimasoni-Florens to compute ρ-invariants. This generalizes results of Cochran-Teichner and Friedl on knots to the setting of links. As an application, we prove with only twelve possible exceptions that the twist knots of algebraic order two are linearly independent in the topological …
Proves Montesinos-Nakanishi 3-move conjecture for links up to 20 crossings.
problem Every link is 3-move equivalent to a trivial link.
method Computational methods, including new code in Regina.
result Proves conjecture for links with up to 20 crossings.
Algorithm for Teichmüller polynomial of certain fibered links.
problem Distinguishing fibered alternating links.
method Algorithm for computing Teichmüller polynomial for fibered alternating links associated with trees.
result Exhibit a mutant pair of links distinguished by Teichmüller polynomial.
Computes entanglement entropy using Chern-Simons theory and symmetric webs.
problem Determining if a product state implies unlinked components.
method Using symmetric webs to compute colored link invariants and write multi-partite entangled states.
result Written down multi-partite entangled states of any given link.
By using the cohomology theory of quandles, quandle cocycle invariants and shadow quandle cocycle invariants are defined for oriented links and surface-links via broken surface diagrams. By using symmetric quandles, symmetric quandle cocycle invariants are also defined for unoriented links and surface-links via broken …