Study satellite knots using bordered Floer theory, proving non-thinness and calculating genus.
problem Properties of twisted Mazur pattern satellite knots.
method Use bordered Floer theory to analyze knots and calculate genus.
result Prove non-thinness of Qn(K) and calculate 3-genus in terms of n and K. An oriented link is positive if it has a link diagram whose crossings are all positive. An oriented link is almost positive if it is not positive and has a link diagram with exactly one negative crossing. It is known that the Rasmussen invariant, 4-genus and 3-genus of a positive knot are equal. In this paper, we p…
The knot concordance invariant Upsilon, recently defined by Ozsvath, Stipsicz, and Szabo, takes values in the group of piecewise linear functions on the closed interval [0,2]. This paper presents a description of one approach to defining Upsilon and of proving its basic properties related to the knot 3-genus, 4-genus, …
The paper proves a conjecture about satellite knots and their slice genus.
problem The topological slice genus of satellite knots and its bounds.
method Establishes the conjecture for a variant of the topological slice genus, the Z-slice genus.
result The topological slice genus of a satellite knot is bounded above by the sum of the slice genera of the knot and the pattern.
Introduces knot theory via surface perspectives.
problem Understanding knots through surface geometry.
method Explains isotopies, Reidemeister moves, and Seifert surfaces.
result Introduces a group structure on knots.
Study shows non-equivariant and equivariant non-orientable 4-genus of periodic knots can differ.
problem Exploring differences in non-orientable 4-genus for periodic knots.
method Analyzed p-periodic knots, showing differences in equivariant and non-equivariant non-orientable 4-genus.
result Differences exist in non-equivariant and equivariant non-orientable 4-genus for periodic knots.
Proves any three or more knots can form a genus-zero link in a 3-manifold.
problem Realizing any finite collection of knots as components of a genus-zero link.
method Proves the realizability of knots as components of genus-zero links in 3-manifolds, controlling pairwise linking numbers.
result Any finite collection of at least three isotopy classes of knots can form a genus-zero link in a 3-manifold, satisfying a specific condition.
The paper calculates a knot invariant for 3-braid knots.
problem Calculating the concordance invariant for 3-braid knots.
method Constructing cobordisms between 3-braid knots and torus knots.
result Explicit formulas and values for the invariant υ(K) for 3-braid knots. Study on nonorientable 4-genus of double twist knots.
problem Determining the nonorientable 4-genus of double twist knots.
method Explicit constructions and obstructions from Donaldson's diagonalization theorem.
result Proved bounds on nonorientable 4-genus for infinite subfamilies of double twist knots.
New bounds on odd multicrossing numbers of knots and links are established.
problem Determining bounds on the number of crossings in knots and links.
method Proved inequalities involving the (2k+1)-crossing number, 3-genus, and number of components of a link. result Established new bounds on the odd crossing numbers of torus knots and links.
One of Powell's generators is not necessary.
problem Unresolved conjecture about generating Goeritz group.
method Short argument showing redundancy of one generator.
result One of Powell's generators is a consequence of others.
Improved bound on first eigenvalue of minimal surfaces in S3.
problem Bounding the first eigenvalue of minimal surfaces in S3. method Proved λ1≥1+εg for embedded minimal surfaces Σ in S3. result Improved bound on the first eigenvalue of λ1. 33 curves on a 3-genus surface, all intersecting at most once.
problem Finding a saturated system of curves on a surface of genus 3.
method Constructing 33 essential curves pairwise non-homotopic and intersecting at most once.
result The constructed system is saturated, not properly contained in any other system.
For all genus g, Powell's elements generate Goeritz groups trivially.
problem Tackles the stability of Powell's Conjecture on Goeritz groups of S^3.
method Shows that the natural function is trivial for each genus g.
result For each genus g, the natural function from G_g to G_{g+1}/P_{g+1} is trivial.
Klein quartic maximizes the first positive Laplacian eigenvalue's multiplicity to 8.
problem Maximizing the first positive eigenvalue's multiplicity of the Laplacian.
method Analyzing hyperbolic surfaces of genus 3 and 2, proving the Klein quartic's maximality.
result Klein quartic maximizes the first positive Laplacian eigenvalue's multiplicity to 8.
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…
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.
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.
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.
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.
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.
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.
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.
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.