Symmetric quandles provide new insights into link colorings.
problem Understanding link colorings using quandles.
method Construction of symmetric quandles and isomorphism of their cohomology groups.
result Homology groups of quandles are isomorphic to those of their symmetric doubles.
Categorifies symmetric link invariants using foam technology.
problem Categorify symmetric link invariants using slN-webs. method Finite dimensional categorification of symmetric evaluation of slN-webs using foam technology. result Categorifies symmetric link invariants, providing a spectral sequence from Khovanov-Rozansky to symmetric.
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.
A quandle coloring obstruction prevents a specific link from being ribbon concordant.
problem Obstructing a specific link from being ribbon concordant.
method Symmetric dihedral quandle coloring analysis.
result A symmetric dihedral quandle of order 4 cannot color a specific surface-link, obstructing ribbon concordance.
Right inverse found for Cartan differential in rank-1 symmetric spaces.
problem Finding a right inverse for the Cartan differential in symmetric spaces.
method Integral operator approach to the Cartan differential on exact forms.
result Extension of Gauss linking integral to rank-1 symmetric spaces.
Characterizes knot groups and symmetric quandles of surface-links.
problem Characterize knot groups and symmetric quandles of surface-links.
method Used plat presentations for surface-links and closed 2-dimensional braids.
result Generalized results to include non-orientable surface-links and showed that dihedral quandles can be realized as symmetric quandles of surface-links.
New divide with gleams method simplifies symmetric link representation.
problem Representing symmetric links with divides and gleams.
method Introducing divides with gleams and algorithms for transvergent diagrams.
result Divides with gleams simplify the representation of symmetric links.
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.
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.
A symmetric matrix invariant is defined for oriented link diagrams.
problem Defining an invariant for oriented link diagrams.
method Defining a symmetric map τD from regions of an oriented link diagram to Z[x], corrected by the writhe. result The negative signature of τD, corrected by the writhe, conjecturally equals twice the Tristram-Levine signature function. The paper defines a preorder on links and explores its implications for symmetric unions.
problem Understanding the relationships between links and their symmetric unions.
method Defining a preorder relation and proving properties of links and their orbifold groups.
result If a link L is a Montesinos link with r≥3 rational tangles, then L′ is either a Montesinos link with at most r+1 rational tangles or a certain connected sum. 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. Study of intrinsic symmetry groups of links, finding counterexamples.
problem Whether every subgroup of the symmetric group is an intrinsic symmetry group of some link.
method Defined intrinsic symmetry groups and provided counterexamples.
result For n > 5, there does not exist an n-component link L for which S(L) is the alternating group.
The paper explores symmetric representations of links and conditions for amphichirality.
problem Investigating symmetric representations of links and conditions for amphichirality.
method Using antipodally self-dual and antipodally symmetric maps, the authors provide sufficient combinatorial conditions for amphichirality.
result A link is amphichiral if its self-dual pairing is not one of 6 specific ones.
The paper shows links can be colored with fewer colors than previously thought.
problem Coloring links using the symmetric group of degree three.
method Analyzing the number of colors for link colorings by S3. result 2-bridge links with 5 colors can be colored with only 4 colors.
We introduce the notion of a ribbon-clasp surface-link, which is a generalization of a ribbon surface-link. We generalize the notion of a normal form on embedded surface-links to the case of immersed surface-links and prove that any (immersed) surface-link can be described in a normal form. It is known that an embedded…
We prove a version of symmetric criticality for ropelength-critical knots. Our theorem implies that a knot or link with a symmetric representative has a ropelength-critical configuration with the same symmetry. We use this to construct new examples of ropelength critical configurations for knots and links which are dif…
We introduce a notion of symmetric Whitney tower cobordism between bordered 3-manifolds, aiming at the study of homology cobordism and link concordance. It is motivated by the symmetric Whitney tower approach to slicing knots and links initiated by Cochran, Orr, and Teichner. We give amenable Cheeger-Gromov rho-invaria…
The paper defines axis systems for link projections and characterizes them for twist knots.
problem Defining and characterizing axis systems for link projections.
method Defined axis systems of link projections and characterized them for twist knots.
result Characterized axis systems of standard projections of twist knots.
Paper introduces symmetric divergence link models for probability distributions.
problem Symmetric divergence measures for probability distributions.
method Two general classes of link models: one for survival functions and another for cumulative probability distribution functions.
result Advantages of symmetric divergence measures over asymmetric measures for model averaging and feature assessment.
Simplified computation of symmetric gl_1 homology for links.
problem Computing the symmetric gl_1 homology for uncolored links.
method Down-to-earth description, basis construction, and algorithm for computation.
result An algorithm and program for computing the invariant for uncolored links.
Symmetric function LM,N lifts torus link homology.
problem Computing the triply-graded Khovanov-Rozansky homology of torus links.
method Defined a symmetric function LM,N and showed it satisfies a recursion for torus link homology. result Triply-graded Khovanov-Rozansky homology of torus links is a specialization of LM,N. Paper defines conditions for projective links in projective 3-space.
problem Characterizing links in projective 3-space.
method Combinatorial conditions and antipodal symmetry.
result Easy condition to prevent alternating projective links.
Elliptic curve governs Hopf linking in symmetric tensegrity.
problem Understanding Hopf linking in A4-symmetric tensegrity.
method Parametrized by points on the elliptic curve 30a2.
result Only one rational point corresponds to a stable configuration.
Half grid diagrams prove every link can be represented by a special type of grid diagram.
problem Representing links using grid diagrams and related invariants.
method Defining half grid diagrams and constructing canonical pairs, proving equivalence to Jones' construction, relating to classical link invariants.
result Established a new method to relate the oriented Thompson index to classical link invariants and provided bounds for knot invariants.
New exact sequence links cohomology, automorphisms, and extensions of symmetric quandles.
problem Understanding the structure of extensions and automorphisms in symmetric quandles.
method Derived a four-term exact sequence relating 1-cocycles, second cohomology, and automorphisms.
result Obstruction to automorphisms lies in the second cohomology of symmetric quandles.
Basket links are shown to be isotopic to T(2,n+1).
problem Understanding isotopy of basket links to torus links.
method Using symmetrized Seifert form congruence to An matrix. result Basket links are isotopic to T(2,n+1). We show that every p-fold strictly-cyclic branched covering of a b-bridge link in the 3-sphere admits a p-symmetric Heegaard splitting of genus g=(b-1)(p-1). This gives a complete converse to a result of Birman and Hilden, and gives an intrinsic characterization of p-symmetric Heegaard splittings as p-fold strictly-cyc…
Study of colored triangulations linked to symmetric groups.
problem Enumeration of permutations up to conjugation.
method Analysis of checker triangulated surfaces and their Belyi data.
result Links between triangulations and infinite symmetric groups.
In this paper, we investigate some applications of commutator subgroups to homotopy groups and geometric groups. In particular, we show that the intersection subgroups of some canonical subgroups in certain link groups modulo their symmetric commutator subgroups are isomorphic to the (higher) homotopy groups. This give…
The paper characterizes links in 3D from divides with cusps.
problem Characterizing links in 3D from divides with cusps.
method Defines and characterizes divides with cusps and their associated links.
result Every strongly invertible link and 2-periodic link can be described as the link of a divide with cusps.
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 …
Develops symmetric Cartan calculus linking to Patterson-Walker metric.
problem No specific problem stated; focuses on developing a new calculus.
method Symmetric Cartan calculus, using torsion-free affine connections.
result Symmetric Cartan calculus is a complete analogue of classical Cartan calculus.
The paper defines conditions for good involutions in generalized Alexander quandles.
problem Determining conditions for good involutions in generalized Alexander quandles.
method Analyzing the structure of generalized Alexander quandles and their involutions.
result Classification of all good involutions in connected generalized Alexander quandles.
Green's functions and Biot-Savart operators on curved spaces quantify linking numbers.
problem Calculating linking numbers on curved spaces.
method Constructing radial fundamental solutions for differential form Laplacian.
result Green's functions and Biot-Savart operators link curved spaces' geometry to linking numbers.
The paper defines new representations and groups related to virtual links.
problem Defining and studying new representations of virtual braid groups and link groups.
method Introducing virtually symmetric representations, virtual link groups, and marked Gauss diagrams.
result Established equivalence of many known representations to virtually symmetric representations.
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.
Using the recently proposed differential hierarchy (Z-expansion) technique, we obtain a general expression for the HOMFLY polynomials in two arbitrary symmetric representations of link families, including Whitehead and Borromean links. Among other things, this allows us to check and confirm the recent conjecture of arX…
We prove that the signature of an even, symmetric form on a finite rank integral lattice, has signature divisible by 8, provided its associated linking form vanishes in the Witt group of linking forms. Our result generalizes the well know fact that an even, unimodular form has signature divisible by 8. We give applicat…
Categorifies symmetric functions and computes invariants of tangles.
problem Computing and characterizing invariants of tangles.
method Using homotopy categories and symmetric monoidal categories.
result Existence of symmetric group actions and spectral sequences.
We show that the triply graded Khovanov-Rozansky homology of the torus link Tn,k stablizes as k→∞. We explicitly compute the stable homology (as a ring), which proves a conjecture of Gorsky-Oblomkov-Rasmussen-Shende. To accomplish this, we construct complexes Pn of Soergel bimodules which categorify t…
We obtain bounds on hyperbolic volume for periodic links and Conway sums of alternating tangles. For links that are Conway sums we also bound the hyperbolic volume in terms of the coefficients of the Jones polynomial.
The elliptic Hall algebra governs torus link homology.
problem Proving the elliptic Hall algebra's role in torus link homology.
method Developed a rational Shareshian-Wachs involution to prove the symmetry of generating functions.
result Resolved a conjecture by establishing the elliptic Hall algebra's role in torus link homology.
Character variety of Whitehead link described in detail.
problem Character variety of Whitehead link.
method Nice description of a Zariski open subset.
result Zariski open subset of the character variety described.
New inequalities link boundary lengths to orthogeodesics in symmetric spaces.
problem Understanding maximal representations in symmetric spaces.
method Proving inequalities for lengths of boundary and orthogeodesics.
result Generalized Basmajian's identity for symmetric spaces.
New theorem links quaternionic-Kähler manifolds to symmetric spaces.
problem Understanding curvature properties of quaternionic-Kähler manifolds.
method Analyzing positive scalar curvature and nonnegative sectional curvature.
result Compact quaternionic-Kähler manifolds with these properties are symmetric.
We introduce the notion of a quandle with a good involution and its homology groups. Carter et al. defined quandle cocycle invariants for oriented links and oriented surface-links. By use of good involutions, quandle cocyle invariants can be defined for links and surface-links which are not necessarily oriented or orie…
We give a concise proof of a classification of lens spaces up to orientation-preserving homeomorphisms. The chief ingredient in our proof is a study of the Alexander polynomial of ` symmetric' links in S3.