New algebraic method for knot Floer homology computation.
problem Computing knot Floer homology efficiently.
method Extending bordered Floer homology to partial knot projections and establishing a pairing result.
result Identification of knot Floer homology with its algebraic definition.
A knot K is called n-adjacent to another knot K', if K admits a projection containing n generalized crossings such that changing any 0 < m \leq n of them yields a projection of K'. We apply techniques from the theory of sutured 3-manifolds, Dehn surgery and the theory of geometric structures of 3-manifolds to answer th…
This paper tabulates prime knot projections up to eight double points.
problem Tabulating prime knot projections and their mirror images up to a certain number of double points.
method Systematic flypes and enumeration of tangles with at most four double points, using arrow diagrams.
result Complete table of prime knot projections with their mirror images up to eight double points.
Legendrian knots can be represented by projections with multi-crossings.
problem Representing Legendrian knots with multi-crossings.
method Investigating übercrossing and petal projections in front and Lagrangian projections.
result Legendrian knots with übercrossing projections in front are smoothly isotopic to the unknot.
An algorithm determines knot colorability and determinants from petal projections.
problem Determining knot colorability and determinants from petal projections.
method Algorithm based on petal projections and permutations.
result Determinants of all prime knots with crossing number less than 10 computed.
New number bounds knot complexity, including unknotting and crosscap numbers.
problem Bounding knot complexity and understanding knot types.
method Introducing an unknotting-type number to estimate crosscap number.
result Determines set of knots with crosscap number at most two.
The study finds lower bounds for the warping degree of a knot projection.
problem Determining the warping degree of a knot projection.
method Examining the maximal number of regions sharing no crossings for a fixed crossing in a knot projection.
result Lower bounds for the warping degree of a knot projection are provided.
The paper defines a new equivalence relation for knot projections and finds an infinite number of distinct classes.
problem Classifying knot projections based on weak homotopy equivalence.
method Defining weak (1, 2, 3) homotopy and using it to find an invariant.
result There are an infinite number of weak (1, 2, 3) homotopy equivalence classes of knot projections.
An increasing sequence of integers is said to be universal for knots if every knot has a reduced regular projection on the sphere such that the number of edges of each complementary face of the projection comes from the given sequence. Adams, Shinjo, and Tanaka have, in a work, shown that (2,4,5) and (3,4,n) (where n i…
Projection maps virtual Legendrian knots to classical ones.
problem Classifying virtual Legendrian knots.
method Developed a projection operation from virtual to classical Legendrian knots.
result Virtual crossing number equals classical crossing number.
The paper studies knots in projective space using virtual link theory.
problem Understanding knots in three-dimensional projective space.
method Associate virtual links to projective links and apply virtual knot theory techniques.
result Equivalent projective links correspond to equivalent virtual links modulo a flype move.
The paper finds petal numbers of torus knots using superbridge indices.
problem Determining petal numbers of torus knots.
method Using superbridge indices, the paper establishes relations between superbridge indices and petal numbers of torus knots.
result The petal number of Tr,s is found to be 2s−1 when 1<r<s and r≡1mods−r. The upper bound is $2s - 2\Big\lfloor \frac{s}{r} \Big
floor +1$. This paper identifies knot projections with reductivity two.
problem Determining knot projections with a specific reductivity level.
method Examined four types of reductivity (Seifert type splice, non-Seifert type splice, recursively, simultaneously) and their combinations.
result Identified all knot projections with reductivity two for the four definitions.
Prove strong ribbon concordance induces a partial order on links, certify minimality for a handful of knots, and find minimal ribbon minimal knots.
problem Prove strong ribbon concordance induces a partial order on links.
method Use results from knot Floer homology to certify minimality under the ribbon partial order.
result Certify minimality for a handful of knots and find minimal ribbon minimal knots.
Let n be any integer greater than two. We prove that there exists a projection P having the following properties. (1) P is not the projection of any unknotted knot. (2) The singular point set of P consists of double points. (3) P is the projection of an n-knot which is diffeomorphic to the standard sphere. We prove the…
The paper defines new homotopy relations on knot projections and classifies certain knot types.
problem Defining and classifying knot homotopy relations.
method Introducing cross chord numbers and using them to define strong and weak (1, 3) homotopies.
result Complete classification of knot projections with trivializing number two.
New homotopy types and invariants defined for knots.
problem Defining and characterizing different homotopy types of knot projections.
method Introducing strong and weak (1, 2) homotopies and defining new invariants.
result New necessary and sufficient conditions for homotopy equivalence of knot projections.
Introduced recently, an n-crossing is a singular point in a projection of a link at which n strands cross such that each strand travels straight through the crossing. We introduce the notion of an übercrossing projection, a knot projection with a single n-crossing. Such a projection is necessarily composed of a collect…
In this paper, we extend the theory of sutured Floer homology developed by the author. We first prove an adjunction inequality, and then define a polytope P(M,g) in H^2(M,\partial M; R) that is spanned by the Spin^c-structures which support non-zero Floer homology groups. If (M,g) --> (M',g') is a taut surface decompos…
Partial proof of a conjecture about knot concordance maps.
problem Proving a conjecture about homomorphisms in knot concordance.
method Analyzing self-maps of the knot concordance group.
result Proved a map is not a homomorphism for certain winding numbers.
Paper proves unique canonical form for certain highly twisted knots and links.
problem Classifying knots and links with specific plat projections.
method Analyzes 2m-plat projections with specific constraints on crossings and heights. result Unique canonical form for certain knots and links with plat projections.
32 knot projections classified based on forbidden Reidemeister moves.
problem Classifying knot projections based on allowed Reidemeister moves.
method 32 homotopy classifications based on forbidden Reidemeister moves.
result 20 non-trivial cases of knot projections are mutually different.
Paper defines weak (1, 3) homotopy for knot projections and classifies trivial knots.
problem Classifying knot projections under weak (1, 3) homotopy.
method Defines weak (1, 3) homotopy, introduces a map to knot isotopy classes, and determines trivial knots.
result Determines which knot projections are trivial under weak (1, 3) homotopy.
A quadruple crossing is a crossing in a projection of a knot or link that has four strands of the knot passing straight through it. A quadruple crossing projection is a projection such that all of the crossings are quadruple crossings. In a previous paper, it was proved that every knot and link has a quadruple crossing…
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.
Knots in Euclidean space which may be parameterized by a single cosine function in each coordinate are called Lissajous knots. We show that twist knots are Lissajous knots if and only if their Arf invariants are zero. We further prove that all 2-bridge knots and all (3,q)-torus knots have Lissajous projections.
New knots found with same determinant but no symmetric relation.
problem Determining if knots with the same determinant are symmetrically related.
method Constructing a family of knots with the same determinant but no symmetric relation.
result No two knots in the family are symmetrically related.
We give the bridge indices for 11-crossing prime knots and give a minimal bridge projection for each of these knots. The results on the indices may be easily summarized: all of these knots that are not rational knots or Montesinos knots have bridge index three.
The paper shows that knot projections without triple chords can be simplified.
problem The study of knot projections and their chord diagrams.
method Flat Reidemeister moves that decrease 1-gons or strong 2-gons.
result For any knot projection without triple chords, a sequence of moves simplifies it to a simple closed curve.
A partial order on the set of prime knots can be defined by the existence of an epimorphism between knot groups. We prove that all the prime knots with up to 6 crossings are minimal. We also show that each fibered knot with the irreducible Alexander polynomial is minimal.
In this paper we propose {\it a region choice problem} for a knot projection. This problem is an integral extension of Shimizu's 'region crossing change unknotting operation.' We show that there exists a solution of the region choice problem for all knot projections.
Defines a measure of knot concordance using cobordism distance.
problem Measuring how close knots are to being linearly dependent.
method Cobordism distance on cyclic subgroups of knot concordance group.
result Projective space of knot concordance group with integer-valued metric.
The paper proves tight fibered knots are minimal in a specific knot order.
problem Understanding the minimality of tight fibered knots.
method Proved ribbon concordance forms a partial order and used it to show tight fibered knots are minimal.
result All tight fibered knots are minimal in the ribbon concordance order.
Proves partial-dual genus polynomial is a knot invariant weight system.
problem Proving the partial-dual genus polynomial is a weight system.
method Proved the polynomial satisfies the four-term relation, thus making it a weight system.
result The partial-dual genus polynomial is a Vassiliev knot invariant weight system.
In this paper we use continued fractions to study a partial order on the set of 2-bridge knots derived from the work of Ohtsuki, Riley, and Sakuma. We establish necessary and sufficient conditions for any set of 2-bridge knots to have an upper bound with respect to the partial order. Moreover, given any 2-bridge knot K…
A triple crossing is a crossing in a projection of a knot or link that has three strands of the knot passing straight through it. A triple crossing projection is a projection such that all of the crossings are triple crossings. We prove that every knot and link has a triple crossing projection and then investigate c_3(…
Ribbon cobordism forms a partial order in 3-manifolds.
problem Understanding partial orders in 3-manifolds.
method Utilizing recent methods from Ian Agol's work on knot concordance.
result Ribbon rational homology cobordism forms a partial order.
The paper classifies knots in real projective 3-space and introduces new geometric tools.
problem Classifying knots in real projective 3-space and understanding their properties.
method Structural theorem, space bending surgery, genus definition, non-cancellation theorem.
result The genus detects knottedness and classifies knots in real projective 3-space.
Study bounds on cusp volumes of alternating knots on surfaces.
problem Bounding cusp volumes of knots on surfaces.
method Analyzing hyperbolic knots with alternating projections on embedded surfaces.
result Two-sided bounds on cusp area in terms of twist number and surface genus.
Knots can be ordered by ribbon concordance, solving a long-standing question.
problem Ordering knots by ribbon concordance.
method Representation varieties of knot groups to SO(N) and relations induced by ribbon concordance. result Ribbon concordance forms a partial ordering on the set of knots.
Two new invariants that are closely related to Milnor's curvature-torsion invariant are introduced. The first, the spiral index of a knot, captures the minimum number of maxima among all knot projections that are free of inflection points. This invariant is closely related to both the bridge and braid index of the knot…
An increasing sequence of integers is said to be universal for knots and links if every knot and link has a projection to the sphere such that the number of edges of each complementary face of the projection comes from the given sequence. This paper is an investigation into which sequences, either finite or infinite, a…
Study jets of flat partial connections in foliations.
problem Characterize and understand flat partial connections in foliations.
method Define and apply jets to flat partial connections in smooth foliations and locally free sheaves, focusing on codimension one and arbitrary codimension foliations.
result Define and apply jets to characterize transversely affine and projective structures in foliations.
Solves if a link projection can represent a specific link.
problem Determining if a link projection represents a specific link L6n1. method Analyzes the link L6n1 and uses previous results for prime knots and links with crossing number ≤ 5. result Proves that L6n1 can be represented by a specific link projection. Studies projective geometry and partial differential equations prolongation.
problem Understanding the prolongation of overdetermined geometric partial differential equations.
method Introduction to differential geometry and tractor calculus, study of prolongation of equations.
result Recovery of projective tractor and cotractor connections via partial differential equations prolongation.
The paper proves rigidity of surgeries on the figure-eight knot complement.
problem Infinitesimal projective rigidity of surgeries on the figure-eight knot complement.
method Computer-assisted proof and explicit representations of the knot complement.
result Proves infinitesimal projective rigidity for surgeries far from the ideal point.
We show that any nontrivial reduced knot projection can be obtained from a trefoil projection by a finite sequence of half-twisted splice operations and their inverses such that the result of each step in the sequence is reduced.
We give a simple example showing that a knot or link diagram that lies in the Z2 lattice is not necessarily the projection of a lattice stick knot or link in the Z3 lattice, and we give a necessary and sufficient condition for when a knot or link diagram that lies in the Z2 lat…