New categorification method for infinite braids.
problem Categorifying highest-weight projectors for infinite braids.
method Limiting colored Khovanov-Rozansky homology of infinite braids.
result Partial isomorphism between HOMFLY-PT homology of braids and infinite torus knots.
Infinite braids' categorification proven using Khovanov homology.
problem Categorifying Jones-Wenzl projectors for infinite braids.
method Limiting Khovanov chain complex and homotopy types.
result Jones-Wenzl projectors categorified for infinite braids.
A sequence of Temperley-Lieb algebra elements corresponding to torus braids with growing twisting numbers converges to the Jones-Wenzl projector. We show that a sequence of categorification complexes of these braids also has a limit which may serve as a categorification of the Jones-Wenzl projector.
Proves method to compute HOMFLYPT skein module of lens spaces L(p,1) from solid torus.
problem Computing HOMFLYPT skein module of lens spaces L(p,1) from solid torus. method Solving infinite system of equations involving Lambropoulou invariant and braid band moves.
result Derives HOMFLYPT skein module of lens spaces L(p,1) from solid torus. New reflection groups derived from torus knots with finite meridians.
problem Understanding reflection groups derived from torus knot groups with finite meridians.
method Using the theory of J-groups and Coxeter groups, study quotients of torus knot groups.
result Classification of toric reflection groups and their properties.
Researchers compute HOMFLYPT skein module of lens spaces using braids.
problem Computing the HOMFLYPT skein module of lens spaces L(p,1). method Using braids to solve an infinite system of equations.
result Established the relation between S(L(p,1)) and S(ST). Researchers compute the skein module of a solid torus using braids.
problem Computing the HOMFLYPT skein module of S1imesS2. method Braid-theoretic techniques to solve an infinite system of equations.
result The free part of the skein module is generated by the empty link, with all other elements being torsion.
Twisted torus links are given by twisting a subset of strands on a closed braid representative of a torus link. T--links are a natural generalization, given by repeated positive twisting. We establish a one-to-one correspondence between positive braid representatives of Lorenz links and T--links, so Lorenz links and T-…
Formula found for braid index of n-bridge braids.
problem Finding a formula for the braid index of n-bridge braids. method Elementary, effective, self-contained proof.
result Closed form formula for braid index of n-bridge braids. Researchers develop a new basis to simplify solving infinite systems for HOMFLYPT skein module of lens spaces.
problem Computing the HOMFLYPT skein module of lens spaces L(p,1) via braids. method Using a new basis Λ of the HOMFLYPT skein module of the solid torus, relating infinite systems of equations obtained by performing braid band moves on elements in Λ+ and Λ− via a map I. result Reduced complexity in solving the infinite system of equations for L(p,1) using braids. We consider a surface link in the 4-space which can be presented by a simple branched covering over the standard torus, which we call a torus-covering link. Torus-covering links include spun T2-knots and turned spun T2-knots. In this paper we braid a torus-covering link over the standard 2-sphere. This gives an u…
This paper studies the normal closure of braid groups in a torus.
problem Understanding the normal closure of braid groups in a torus.
method Combining results from Birman and Goldberg, the paper explores the geometric interpretation of the normal closure of full braid groups.
result The normal closure of Bn(D) in Bn(T) has a geometric description. New Garside structures found for torus knot groups and related braid groups.
problem Finding Garside structures for torus knot groups and related braid groups.
method Introducing a new Garside monoid M(n,m) for (n,m)-torus knot groups and other braid groups. result New Garside structures for (n,m)-torus knot groups and related braid groups are constructed. New link groups are derived from torus necklaces, connecting braid groups to reflection groups.
problem Understanding the relationship between braid groups and reflection groups.
method Constructing torus necklaces and linking them to braid groups of J-reflection groups. result Link groups of torus necklaces are precisely braid groups of J-reflection groups, with meridians as braid reflections. This paper computes the Homflypt skein module of lens spaces using braids.
problem Computing the Homflypt skein module of lens spaces L(p,1). method Using Lambropoulou invariant and braid band moves on a basis of the solid torus.
result Established the connection and computed the Homflypt skein module of lens spaces.
We simplify Khovanov homology for torus braids using Gaussian elimination.
problem Computing Khovanov homology for torus braids is complex and computationally intensive.
method Applying Gaussian elimination to reduce the number of generators in the Khovanov chain complex.
result We provide a bound on the number of generators in the whittled complex at fixed homological degree.
A new Markov theorem for 4D ribbon torus-links.
problem Describing isotopic links in R4. method Develops a theorem for ribbon torus-links in B3imesS1. result First step towards a 4D Markov theorem.
Study links' arc index and Turaev genus, proving conjectures.
problem Understanding the arc index and Turaev genus of links.
method Computed arc index, established bounds, and conjectured inequalities.
result Proved conjectures linking crossing number, arc index, and Turaev genus.
Study cobordism distances between 3-braid links and trefoil knots.
problem Understanding the geometric relationship between 3-braid links and trefoil knots.
method Determined cobordism distances between 3-braid links and trefoil knots, and explored limits of Coxeter's braid group result.
result Found cobordism distances between 3-braid links and trefoil knots, up to a constant error.
Study fixed points of n-valued maps on surfaces using braid groups.
problem Deciding when n-valued maps can be deformed to free of fixed points.
method Fixed point theory of maps between X and its configuration spaces, braid groups.
result Algebraic criterion for surfaces to determine fixed point free n-valued maps.
Alternative basis for Kauffman bracket skein module of solid torus found using braids.
problem Computing Kauffman bracket skein module of lens spaces.
method Using Temperley--Lieb algebra of type B and braids.
result Alternative basis BmST for ${
m KBSM}\left({
m ST}
ight)$. Extends Khovanov cohomology to colored links using stable homotopy types.
problem Extending Khovanov cohomology to colored links.
method Defining a stable homotopy type X_col(L_c) for colored links L, using categorified Jones-Wenzl projectors and infinite torus braids.
result Computes stable homotopy types for specific colored links and makes a conjecture for others.
To a closed braid in a solid torus we associate a trace graph in a thickened torus in such a way that closed braids are isotopic if and only if their trace graphs can be related by trihedral and tetraherdal moves. For closed braids with a fixed number of strands, we recognize trace graphs up to isotopy and trihedral mo…
New homology for infinite multi-colored braids, completing previous work.
problem Categorification of highest-weight projectors for infinite braids.
method Defining limiting Khovanov-Rozansky homology for semi-infinite braids.
result Categorifies highest-weight projectors for a large class of braids.
The paper computes the Kauffman bracket skein module of S1imesS2 via braids.
problem Computing the Kauffman bracket skein module of S1imesS2. method Two methods: extending a universal invariant to S1imesS2 via braid band moves and a diagrammatic approach. result The Kauffman bracket skein module of S1imesS2 is not torsion-free and its free part is generated by the unknot. The paper proves that certain thin links have only Z2 torsion in Khovanov homology.
problem Understanding torsion in Khovanov homology for thin links.
method Local analysis of Khovanov homology for links supported on two adjacent diagonals.
result Khovanov homology of thin links has only Z2 torsion over a specified range of homological gradings.
We calculate the alternating number of torus knots with braid index 4 and less. For the lower bound, we use the upsilon-invariant recently introduced by Ozsváth, Stipsicz, and Szabó. For the upper bound, we use a known bound for braid index 3 and a new bound for braid index 4. Both bounds coincide, so that we obtai…
Construct petal diagrams from simple braids to verify knot petal numbers.
problem Verify the petal number conjecture for nontrivial torus knots.
method Construct petal diagrams from simple braids and apply to torus knots.
result Confirm petal number conjecture for torus knots, deducing specific conditions.
Explicit formula found for torus knot homology.
problem Computing the homology of torus knots.
method Elias-Hogancamp method and combinatorics of toric braids.
result Explicit formula for triply graded Khovanov-Rozansky homology of torus knots.
Study shows surprising cobordism distances between certain torus knots.
problem Determining cobordism distances between thin and thick torus knots.
method Analyzes locally flat cobordisms between torus knots with small and large braid indices.
result Surprising fact about torus knots as cross-sections of almost minimal cobordisms.
New examples of Legendrian links with infinitely many fillings.
problem Understanding the structure of Legendrian links and their fillings.
method New combinatorial formula for Legendrian contact DGAs and Floer-theoretic techniques.
result Construction of the first families of Legendrian links with infinitely many Lagrangian fillings.
The paper studies which branched covers can be lifted to braided embeddings.
problem Which branched covers lift to braided embeddings?
method Analyzes conditions for liftability using Hansen's criterion and examples in different dimensions.
result Not all branched covers lift to braided embeddings; examples are provided in various dimensions.
New polynomial invariants detect non-invertibility of closed braids.
problem Detecting non-invertibility of closed braids.
method Polynomial 1-cocycles evaluated on closed braids' full rotation.
result Polynomial invariants can detect non-invertibility of 2-component links.
Positive braids have endless filling possibilities.
problem Understanding exact Lagrangian fillings of positive braid links.
method Analyzing positive braid Legendrian links and their fillings.
result Positive braid Legendrian links have infinitely many exact Lagrangian fillings.
New method for computing Kauffman bracket skein module of lens spaces using unoriented braids.
problem Computing Kauffman bracket skein module of lens spaces L(p,q) for qeq0. method Developed a braid theoretic approach via unoriented braids, introducing a new algebra and invariant.
result Computed the Kauffman bracket skein module of lens spaces L(p,1) and extended to q>1. Abstract: Generalized full twist formula for superpolynomials of torus knots.
problem Finding a formula for superpolynomials of torus knots.
method Using Mellit's results, generalized full twist formula for superpolynomials of positive toric braids.
result Generalized full twist formula for superpolynomials of torus knots.
Discrete Morse theory simplifies Khovanov homology calculations.
problem Calculating Khovanov homology using traditional methods is complex.
method Employing discrete Morse theory for knot homologies.
result Advances understanding of Khovanov homology of 3-braids.
The study connects twist positivity to L-space knots and concordance.
problem Understanding the relationship between twist positivity and knot concordance.
method Analyzing Alexander polynomials and braid indices of knots.
result For twist positive L-space knots, the braid index equals the bridge index.
The Upsilon invariant bounds cobordisms between knots and their braid index.
problem Bounding cobordisms between knots and their braid index.
method Using Ozsváth, Stipsicz, and Szabó's Upsilon-invariant.
result Established inductive formulas for the Upsilon invariant of torus knots.
New move proves infinitely many non-conjugate braids for certain links.
problem Proving infinitely many non-conjugate braids for certain link types.
method Introducing a non-degenerate exchange move.
result Links have infinitely many non-conjugate braids after applying the move.
A knot type is exchange reducible if an arbitrary closed n-braid representative can be changed to a closed braid of minimum braid index by a finite sequence of braid isotopies, exchange moves and +/- destabilizations. In the manuscript [J Birman and NC Wrinkle, On transversally simple knots, preprint (1999)] a transver…
Paper computes skein modules of 3-manifolds using braids.
problem Computing skein modules of 3-manifolds.
method Using braids and knot algebras.
result Braid approach to HOMFLYPT and Kauffman bracket skein modules of specific 3-manifolds.
Paper constructs infinitely many non-braid positive hyperbolic L-space knots.
problem Proving the non-braid positivity of hyperbolic L-space knots.
method Constructs infinitely many hyperbolic L-space knots that are not braid positive.
result Distinct examples from Baker and Kegel's constructions.
Study of infinite-string braids as a limit of finite ones.
problem Properties of infinite-string braids.
method Investigate via direct limit of finite n-braid groups.
result Basic properties of countable-string braids.
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. We introduce a new construction of a surface link in the 4-space. We construct a surface link as a branched covering over the standard torus, which we call a torus-covering link. We show that a certain torus-covering T2-link is equivalent to the split union of spun T2-links and turned spun T2-links. We show th…
Let φ:S1×D2→S1 be the natural projection. An oriented knot K↪V=S1×D2 is called an almost closed braid if the restriction of φ to K has exactly two (non-degenerate) critical points (and K is a closed braid if the restriction of φ has no critical points at all). We introduce …
The paper finds braid representatives minimizing simple walks for knots.
problem Finding efficient braid representatives for knots.
method Developed methods to minimize the number of simple walks in braids.
result Computed the colored Jones polynomial for specific knots.