This paper computes the second quandle homology group of knot n-quandles.
problem Characterizing knots using quandle homology groups.
method Computation of the second quandle homology group for knot n-quandles.
result The second quandle homology group of knot n-quandles provides more information than the knot quandle's homology group.
Classifies spatial graphs with finite N-quandles.
problem Determining isomorphism of spatial graphs' N-quandles.
method Generalized N-quandles to spatial graphs, proving basic results and conjecturing a classification.
result Verifies conjecture in several cases, presents a possible counterexample.
Classifies links with finite N-quandles for some N.
problem Describing finite quotients of quandles, especially n-quandles. method Introducing and studying N-quandles, proving one direction of a conjectured classification. result Proves one direction of a conjectured classification of links with finite N-quandles. Complete classification of links and spatial graphs with finite N-quandles.
problem Classifying links and spatial graphs with finite N-quandles.
method Extending fundamental quandle relationships to N-quandles of links and spatial graphs.
result Complete list of links and partial list of spatial graphs with finite N-quandles.
We compute Cayley graphs and automorphism groups for all finite n-quandles of two-bridge and torus knots and links, as well as torus links with an axis.
Researchers describe Cayley graphs for finite involutory quandles of certain two-bridge links.
problem Understanding finite quotients of quandles for knot and link analysis.
method Explicit description of Cayley graphs for finite involutory quandles.
result Explicit descriptions of Cayley graphs for finite involutory quandles of two-bridge links with an axis.
The fundamental n-quandles of links are residually finite for n ≥ 2.
problem Residual finiteness of fundamental n-quandles of oriented links.
method Investigation of residual finiteness and subquandle separability of quandles; use of Winker's work on 3-sphere branched covers.
result Fundamental n-quandles of oriented links are residually finite for each n ≥ 2.
We prove a conjecture of Przytycki which asserts that the n-quandle of a link L in the 3-sphere is finite if and only if the fundamental group of the n-fold cyclic branched cover of the 3-sphere, branched over L, is finite.
Given a presentation for a rack R, we define a process which systematically enumerates the elements of R. The process is modeled on the systematic enumeration of cosets first given by Todd and Coxeter. This generalizes and improves the diagramming method for n-quandles introduced by Winker. We p…
The paper characterizes L-space 3-manifolds using quandle orderability.
problem Characterizing L-space 3-manifolds using quandle orderability.
method Using quandles and their extensions, the paper investigates orderability and circular orderability conditions.
result The n-quandle Qn(L) of a link quandle is not right circularly orderable. Two quandles from Coxeter groups studied, showing similarities in automorphism groups.
problem Understanding quandles from Coxeter groups and their automorphism groups.
method Defined two families of quandles from Coxeter quandles and Pin groups, analyzed their automorphism groups.
result Similarity in inner automorphism groups of the two quandles.
The paper studies finiteness of canonical quotients in Dehn quandles of surfaces.
problem Finiteness of canonical quotients in Dehn quandles of surfaces.
method Analyzing the Dehn quandle structure and using quandle quotients.
result For surfaces of positive genus, the n-quandle of their Dehn quandles is finite for specific values of n. 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.
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.
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.
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 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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Algorithm calculates knot Floer homology for a specific knot type.
problem Computing knot Floer homology for (1,1) knots. method Algorithm based on fundamental group of (1,1) knots. result Algorithm successfully computes knot Floer homology.
Researchers confirm a relation between knot invariants and provide formulas for torus knots.
problem Confirming a relation between knot invariants and providing formulas.
method Explicit formulas and algorithms for certain ADO-invariants of torus knots obtained from the series invariant of knot complements.
result Explicit formulas and algorithms for certain ADO-invariants of torus knots.
Rectangular mosaics extend virtual knot studies to larger polygons.
problem Studying virtual knots using mosaic techniques.
method Introduced rectangular mosaics, modified mosaic moves, and provided invariants.
result Developed algorithms for computing virtual knot invariants.
Classifies 85 tie knots into mathematical categories.
problem Classifying and understanding the mathematical properties of tie knots.
method Formal language and sequence of moves to describe tie knots, classification based on knot theory.
result Proves that any tie knot is prime and alternating.
In a previous paper, we introduced special types of fusions, so called simple-ribbon fusions on links. A knot obtained from the trivial knot by a finite sequence of simple-ribbon fusions is called a simple-ribbon knot. Every ribbon knot with <10 crossings is a simple-ribbon knot. In this paper, we give a formula for th…
We have a knot quandle and a fundamental class as invariants for a surface-knot. These invariants can be defined for a classical knot in a similar way, and it is known that the pair of them is a complete invariant for classical knots. In this paper, we compare a situation in surface-knot theory with that in classical k…
A Lissajous knot is one that can be parameterized by a single cosine function in each coordinate. Lissajous knots are highly symmetric, and for this reason, not all knots are Lissajous. We prove several theorems which allow us to place bounds on the number of Lissajous knot types with given frequencies and to efficient…
Algorithm computes knot Floer complex for knots of thickness one.
problem Computing knot Floer complexes for knots of thickness one.
method Developed and implemented an algorithm for knots of thickness one.
result Algorithm can compute full knot Floer complex for knots of thickness one.