DAHA and skein algebra on a double-torus knot are studied.
problem Understanding the relationship between DAHA and skein algebra on a double-torus surface.
method Combining DAHA of A1-type and CC1-type with the skein algebra on a 4-punctured sphere, constructing DAHA representation for a genus-two surface, and proposing a DAHA polynomial for a double-torus knot.
result A DAHA polynomial for a double-torus knot is proposed, and its relationship with the colored Jones polynomial is discussed.
Generalizes Hecke algebra for double torus, linking to skein algebra.
problem Understanding algebraic structures on double torus.
method Introducing Heegaard dual operators and Dehn twists.
result Established relationship between Hecke algebra and skein algebra.
Researchers found all embeddings of Kuratowski graphs on a double torus.
problem Characterizing embeddings of Kuratowski graphs K3,3 and K5 on the double torus. method Constructive approach using Burnside's Lemma and automorphism groups.
result 14 orientable and 17 non-orientable 2-cell embeddings of K5 on the double torus. Describes curvature lines on a double torus in 4D space.
problem Analyzing curvature lines on a complex geometric shape.
method Using polynomial gradient and Milnor fibration to define curvature lines, then projecting them into 3D space.
result Complete description of curvature lines on the double torus.
This paper classifies semi-equivelar gems on a double torus.
problem Classifying semi-equivelar gems on surfaces with negative Euler characteristic.
method Regular colored graphs representing the double torus, with identical cyclic face degree sequences around each vertex.
result 31 types of semi-equivelar gems on the double torus.
Semi-Equivelar maps are generalizations of Archimedean solids to the surfaces other than 2-sphere. In earlier work a complete classification of semi-equivelar map of type (35,4) on the surface of Euler characteristic -1 was given. In the meantime Karabas an Nedela classified vertex transitive semi-equivelar maps on…
A connected combinatorial 2-manifold is called degree-regular if each of its vertices have the same degree. A connected combinatorial 2-manifold is called weakly regular if it has a vertex-transitive automorphism group. Clearly, a weakly regular combinatorial 2-manifold is degree-regular and a degree-regular combinator…
Semi-Equivelar maps are generalizations of Archimedean Solids (as are equivelar maps of the Platonic solids) to the surfaces other than 2−Sphere. We classify some semi equivelar maps on surface of Euler characteristic -1 and show that none of these are vertex transitive. We establish existence of 12-covered triangula…
Study of hyperbolic polyhedral surfaces with regular faces.
problem Understanding the properties of hyperbolic polyhedral surfaces with regular faces.
method Combinatorial and geometric analysis of hyperbolic polyhedral surfaces with regular faces.
result There is a gap between the areas of non-smooth hyperbolic polyhedral surfaces and smooth hyperbolic surfaces.
Polynomially parametrize interesting knotted surfaces.
problem Constructing polynomial parametrizations of knotted surfaces.
method Develop polynomial parametrization methods for specific knotted surfaces.
result Examples of polynomial parametrizations for knotted spheres, tori, and planes.
New 2-knots found with same knot group but different quandles.
problem Identifying 2-knots with identical knot groups but distinct quandles.
method Analyzing knot quandles of twist spins.
result First example of 2-knots with same knot group but different quandles.
Knots formed from torus knots are not concordant to L-space knots.
problem Understanding concordance in knots formed from connected sums of torus knots.
method Analyzing properties of knots formed from connected sums of torus knots.
result Knots formed from torus knots are not concordant to L-space knots.
New knot quandles distinguish ribbon knots with isomorphic groups.
problem Distinguishing knots with isomorphic fundamental groups.
method Examined knot quandles of Suciu's ribbon knots and computed their types.
result Knot quandles of Suciu's ribbon knots are mutually non-isomorphic.
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.
Study shows differences of torus knots do not form L-space knots.
problem Understanding concordance of knots and their relation to L-space knots. method Examined differences of torus knots and their concordance properties.
result Subgroups generated by two positive torus knots contain no nontrivial L-space knots. Knot contact homology is an invariant of knots derived from Legendrian contact homology which has numerous connections to the knot group. We use basic properties of knot groups to prove that knot contact homology detects every torus knot. Further, if the knot contact homology of a knot is isomorphic to that of a cable …
The paper explores non-trivial welded knots and ribbon torus-knots, proving their existence.
problem Proving the non-triviality of welded knots and ribbon torus-knots.
method By generating examples and determining the fundamental group of the concerned welded knot.
result Non-triviality of welded knots and ribbon torus-knots is demonstrated.
We study Legendrian knots in a cabled knot type. Specifically, given a topological knot type K, we analyze the Legendrian knots in knot types obtained from K by cabling, in terms of Legendrian knots in the knot type K. As a corollary of this analysis, we show that the (2,3)-cable of the (2,3)-torus knot is not transver…
Study concordance of alternating torus knots to L-space knots.
problem When are linear combinations of alternating torus knots concordant to L-space knots?
method Proved Allen's conjecture for alternating torus knots and established a necessary condition.
result Linear combinations of alternating torus knots are concordant to L-space knots if and only if they are a single torus knot.
The paper identifies minimal two-bridge knots with specific conditions.
problem Determining minimal two-bridge knots under certain conditions.
method Analyzing knot groups and epimorphisms.
result Identifies minimal two-bridge knots for specific values of p and q. The paper classifies a special family of knots in lens spaces using knot Floer homology.
problem Classifying constrained knots in lens spaces.
method Parameterization by five integers, characterization via spinc structures, and knot Floer homology calculations. result Complete classification of constrained knots based on knot Floer homology.
This paper studies how knots combine using Alexander Polynomials.
problem How knots combine and their determinants behave.
method Basic knot theory, Alexander Polynomials, and composition techniques.
result Generalized solution for knot determinants in compositions.
The study confirms conjectures about slopes of knots using knot Floer homology.
problem Verifying conjectures about non-integer characterizing slopes of knots.
method Using knot Floer homology, the study verifies conjectures for specific classes of knots.
result Almost all slopes are characterizing for many knots, and infinitely many for L-space knots. Defines slice depth for 2-knots and sets upper bounds for specific knots.
problem Determining the minimum dimension for a 2-knot to be slice.
method Introduces slice depth, defines it for 2-knots, and provides upper bounds for specific knot types.
result Upper bounds for slice depth of certain 2-knots.
Study shows fibered knots' quandles have consistent structure.
problem Understanding the structure of fibered knots through their quandles.
method Analyzes the fiber bundle structure of fibered knots and applies it to their quandles.
result Finiteness and equivalence of knot quandles of fibered knots are discussed.
New diagonal knots found with non-torus structure.
problem Identifying knots with diagonal grid diagrams.
method Analysis of knots represented by diagonal grid diagrams.
result All diagonal knots are positive, and a new non-torus example is found.
New hyperbolic knots not concordant to algebraic ones found.
problem Identifying knots not concordant to algebraic knots.
method Constructing hyperbolic L-space knots.
result Found hyperbolic knots that are not concordant to algebraic knots.
Formula for Alexander polynomial of twisted torus knots derived.
problem Calculating Alexander polynomial for a specific class of knots.
method Knot group presentation combined with Fox's calculus.
result Explicit formula for Alexander polynomial of twisted torus knots.
Knot modules are linked to ribbon 2-knots with geometric constraints.
problem Characterizing knot modules of ribbon 2-knots.
method Showing every Z-torsion free knot module can be realized by a ribbon 2-knot with specific geometric properties. result Knot modules of ribbon 2-knots have geometric constraints with group of geometric dimension at most 2.
Formula for Alexander polynomials of simple-ribbon knots derived.
problem Determining Alexander polynomials for simple-ribbon knots.
method Introduced simple-ribbon fusions and used them to derive a formula for Alexander polynomials.
result Formula for Alexander polynomials of simple-ribbon knots derived.
The paper conjectures Khovanov homology can distinguish torus and twist knots.
problem Detecting and distinguishing knots using Khovanov homology.
method Examining all prime knots with up to 20 crossings, conjecturing Legendrian simplicity.
result Numerical evidence supports Khovanov homology distinguishing torus and twist knots.
Study reveals weak knotting in confined polymers, not dominated by any single knot type.
problem Characterizing knotting in open, confined polymers.
method Modeling open curves as virtual knots, comparing lattice walks and ideal chains in confined and unconfined conditions.
result Weak knotting is a common feature in confined polymers, not dominated by any single knot type.
A quadrisecant of a knot is a straight line intersecting the knot at four points. If a knot has finitely many quadrisecants, one can replace each subarc between two adjacent secant points by the line segment between them to get the quadrisecant approximation of the original knot. It was conjectured that the quadrisecan…
Expanded Legendrian knot atlas for 10-arc index knots.
problem Lack of Legendrian knot data for knots with high arc index.
method Created an atlas of Legendrian knots up to arc index 10.
result Legendrian knots of arc index 10 have been cataloged.
Study grid homology of diagonal knots, finding key terms related to prime factors and decompositions.
problem Determine grid homology of diagonal knots and compare them to other knot types.
method Use grid diagrams and combinatorial knot Floer homology to analyze diagonal knots.
result Grid homology detects the number of prime factors and decompositions of the knot into non-integer tangles.
Two complete knot invariants from diagrams, finite or infinite.
problem Classifying knots completely.
method Constructed two invariants from knot diagrams, finite or infinite.
result Finite set reveals knotting number.
New spectral sequences define knot invariants.
problem Understanding strongly invertible knots.
method Two spectral sequences in knot Floer homology.
result Numerical invariant defined for strongly invertible knots.
Knot concordance linked to involutive knot Floer homology equivalence.
problem Understanding knot concordance through algebraic topology.
method Involutive knot Floer homology and stable equivalence.
result Concordant knots have equivalent involutive knot Floer complexes.
Polynomially parameterizes knots and spheres, proving analogous results.
problem Parameterizing knots and spheres using polynomials.
method Analogous to classical knots, parameterized long 2-knots and certain classes of knotted spheres.
result Polynomial parameterizations for knotted spheres constructed.
New infinite families of twisted torus knots found.
problem Identifying new types of twisted torus knots.
method Finding new infinite families of twisted torus knots with a single negative twist.
result Eight new infinite families of twisted torus knots are discovered.
New knot concept extends welded knots, simplifying classification.
problem Classifying welded knots and their complements.
method Introducing 'wen knots', proving subset relationships, characterizing complements.
result Extended welded knots can be fully characterized by the parity of wens.
Study on random knot diagrams and their probability of forming specific knots.
problem Understanding the probability of forming specific knots from random knot diagrams.
method Analyzing free knot diagrams without over/under information and proving trefoil formation; making conjectures about unknot and trefoil probabilities.
result Every free knot diagram produces trefoil knots, and certain families of diagrams are completely worked out.
Every shadow can be a knot or a connected sum of trefoils.
problem Understanding the structure of shadows and their relation to knots.
method Analyzing reduced shadows and their relationship to specific knots.
result Every shadow corresponds to a torus knot, twist knot, or connected sum of trefoils.
We define cylinder knots as billiard knots in a cylinder. We present a necessary condition for cylinder knots: after dividing cylinder knots by possible rotational symmetries we obtain ribbon knots. We obtain an upper bound for the number of cylinder knots with two fixed parameters (out of three). In addition we prove …
Two-bridge ribbon knots have symmetric union presentations.
problem Characterizing two-bridge ribbon knots.
method Symmetric union presentations and partial knot analysis.
result Symmetric union presentations for various two-bridge ribbon knots.
This paper determines nonhyperbolicity conditions for P/P and P/SF knots.
problem Classifying hyperbolic P/P and P/SF knots.
method Providing necessary, sufficient, or equivalent conditions for nonhyperbolicity.
result Necessary, sufficient, or equivalent conditions for P/P or P/SF knots being nonhyperbolic.
The paper discusses knot colorings and their invariants using Goeritz matrices.
problem Distinguishing knots using coloring methods.
method Elementary approach to equivalence between coloring and Goeritz matrices.
result Computing knot determinant and nullity of pretzel knots.
The paper defines a new knot relation and explores fertile knots with many descendants.
problem Understanding the structure and properties of knot descendants.
method Introducing and studying the descendant relation between knots, focusing on fertile knots.
result Fertile knots have a large number of descendants.