Although most knots are nonalternating, modern research in knot theory seems to focus on alternating knots. We consider here nonalternating knots and their properties. Specifically, we show certain classes of knots have nontrivial Jones polynomials.
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48 results for “nonalternating knots”
Study shows mutation invariance of arc index for some Montesinos knots.
problem Investigate mutation invariance of arc index for nonalternating knots.
method Analyzed families of Montesinos knots, found mutant pairs/triples with same arc index.
result Found infinitely many mutant pairs/triples of Montesinos knots with the same arc index.
It is known that the arc index of alternating knots is the minimal crossing number plus two and the arc index of prime nonalternating knots is less than or equal to the minimal crossing number. We study some cases when the arc index is strictly less than the minimal crossing number. We also give minimal grid diagrams o…
Minimal grid diagrams found for 13-crossing prime knots with 13 arc index.
problem Finding minimal grid diagrams for prime knots with specific crossing and arc indices.
method Used Knotscape to generate spanning trees and obtain minimal arc presentations in grid diagrams.
result 9,988 prime knots with 13 crossings and 13 arc index were identified.
An atlas of Legendrian knotsmath.SG
We present an atlas of Legendrian knots in standard contact three-space. This gives a conjectural Legendrian classification for all knots with arc index at most 9, including alternating knots through 7 crossings and nonalternating knots through 9 crossings. Our method involves a computer search of grid diagrams and app…
The paper converts nonalternating forms of rational links into all-even forms and derives formulas for their braid index and HOMFLY polynomial.
problem Finding formulas for the braid index and HOMFLY polynomial of rational links.
method Algorithm to transform nonalternating forms into all-even forms and derivation of formulas.
result Formulas for the braid index and HOMFLY polynomial of rational links in terms of their reduced alternating form.
Study on random alternating link diagrams and their hyperbolic volumes.
problem Understanding the relationship between the combinatorial structure and hyperbolic volume of random links.
method Model based on random 4-valent maps, analyzing alternating and nonalternating diagrams.
result Expected hyperbolic volume is asymptotically linear in the number of crossings for random alternating diagrams.
New method calculates Thurston norm for 3-manifolds with toroidal boundaries.
problem Computing the Thurston norm for 3-manifolds with toroidal boundaries.
method maw dual graph construction and sutured manifold hierarchies.
result Explicit procedure to compute Thurston norm from hierarchies.
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 -space knots.
problem Understanding concordance of knots and their relation to -space knots.
method Examined differences of torus knots and their concordance properties.
result Subgroups generated by two positive torus knots contain no nontrivial -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 …
Study disproves conjecture about knot approximations.
problem Conjecture about knot approximations was disproved.
method Examined various knot types to find counterexamples.
result Found knots where quadrisecant approximations have different types.
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.
Maps knots to 2-knots, extending braid theory.
problem No specific problem stated; maps knots to 2-knots.
method Constructs a map from knots to 2-knots, extending braid theory.
result Maps knots to 2-knots, extending braid theory.
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 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 spin structures, and knot Floer homology calculations.
result Complete classification of constrained knots based on knot Floer homology.
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 and .
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 -space knots.
Slice depth of 2-knotsmath.GT
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.
Classical knots embed into virtual knots' concordance group.
problem Understanding concordance of virtual knots.
method Proved equivalence between classical and virtual sliceness.
result Embedding of classical knots' concordance group into virtual knots.
Characterizes almost alternating knots from generalizations of alternating knots.
problem Characterizing almost alternating knots.
method Topological characterization based on generalizations of alternating knots.
result A topological characterization of almost alternating knots.
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 -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.
Conjectures on the Khovanov Homology of Torus Knots, Twist Knots, and Legendrian Simple Knotsmath.GT
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.
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 mosaic tabulationmath.GT
Researchers tabulate mosaic numbers of prime knots up to eight crossings.
problem Tabulating mosaic numbers for prime knots up to eight crossings.
method Using the knot mosaic system introduced by Lomonaco and Kauffman, they determined mosaic numbers for all eight-crossing or fewer prime knots.
result Determined mosaic numbers for all eight-crossing or fewer prime 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.
On wen knotsmath.GT
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.
Combinatorial Random Knotsmath.GT
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.
Billiard knots in a cylindermath.GT
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 …
On two-bridge ribbon knotsmath.GT
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.