Paper gives a full-twist inequality for a knot concordance invariant ν^+.
problem Understanding the behavior of the ν^+ invariant under knot cobordisms.
method Developed a full-twist inequality for the ν^+ invariant and extended Wu's cabling formula.
result Extended Wu's cabling formula to all cables and introduced a partial order on ν^+ equivalence classes.
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.
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.
Authors conjecture categorically diagonalizable complex for full twists.
problem Categorically diagonalizing the complex of Soergel bimodules for full twists.
method Utilizes categorical diagonalization theory.
result Proves conjecture in type A, categorifies Young idempotents.
Formula for HOMFLY polynomial in link diagrams.
problem Calculating HOMFLY polynomial for link diagrams.
method Introduced a class of link diagrams, including closed braids, and proved a generalized full-twist formula.
result Generalized formula for HOMFLY polynomial in new class of link diagrams.
New satellite knots found that can't be represented by positive braids with full twists.
problem Satellite knots that cannot be represented by positive braids with full twists.
method Analyzing positive minimal braids and their closures to find satellite knots.
result Infinitely many satellite knots with Lorenz patterns and companions are not Lorenz knots.
Satellite links with many twists have simpler companions.
problem Relationship between satellite and companion links' complexity.
method Constructing satellite links with multiple full twists and analyzing their companion links.
result Satellite links with many twists have simpler companion links.
Characterizes fractional Dehn twist coefficient and proves slice-Bennequin inequality.
problem Understanding the fractional Dehn twist coefficient and its relation to smooth slice genus.
method Characterization of FDTC and establishing the slice-Bennequin inequality.
result Affine linear lower bound for smooth slice genus in terms of FDTC.
New colored knot Floer homology defined using infinite full twists.
problem Defining a new homology theory for knots.
method Defining colored knot Floer homology through colimit of link Floer homology with infinite full twists.
result Colored knot Floer homology is a module over the colored knot Floer homology of the unknot.
Knot invariants and quiver stability linked through full twists.
problem Relating knot invariants to quiver stability under twists.
method Using HOMFLY-PT skein relations and linking/unlinking operations on symmetric quivers.
result Full twists on knots correspond to unlinking or linking of augmented symmetric quivers, confirming stable growth in both.
Formula for Alexander polynomial of links with twists.
problem Computing Alexander polynomial of links with twists.
method Using vector space representation of Uq(gl(1∣1)). result Alexander polynomials stabilize after adding enough twists.
New findings on T-links derived from torus links.
problem Difficulty in determining geometric type of T-links.
method Examined T-links obtained by full twists along torus links.
result T-links obtained by full twists are not torus links.
Positive braids with at least two twists form hyperbolic knots.
problem Classifying knots formed by specific braids.
method Conditions on positive braids with at least two full twists.
result Closure of such braids forms hyperbolic knots.
Satellite links of fully positive braids are characterized.
problem Characterizing satellite links of fully positive braids.
method Analyzing fully positive braids and their satellites.
result Satellite links of fully positive braids are characterized by specific conditions.
New families of twisted torus knots found with essential surfaces.
problem Whether all twisted torus knots have essential tori.
method Analyzing sequences of twists on torus knots.
result Found two new families of toroidal twisted torus knots.
Study of Lorenz links and T-links, showing equivalence and unique presentations.
problem Understanding equivalence and uniqueness of T-link presentations for Lorenz links.
method Minimal-braid approach, V-link braids, T-link parameters, geometric type criteria.
result Explicit correspondence between V-links and T-links, criteria for equivalence and uniqueness.
New combinatorial method connects knot invariants to reflection groups.
problem Computing knot invariants using combinatorial techniques.
method Relating dual braid group generators, Hecke images of pure braids, and reflection groups.
result The (a,z=0)-HOMFLYPT polynomial can be computed as a solution to factorization problems. The paper classifies hyperbolic and satellite T-links formed by twisting.
problem Classifying hyperbolic and satellite T-links formed by twisting.
method Classification through Dehn filling and analysis of parent links.
result Classification of hyperbolic and satellite T-links formed by twisting.
Knot Floer homology stabilizes with twists.
problem Understanding knot behavior through homology.
method Analyzing m full twists on n parallel strands. result Knot Floer homologies stabilize as twists increase.
We prove that the property of admitting no cosmetic crossing changes is preserved under the operation of forming certain satellites of winding number zero. We also define strongly cosmetic crossing changes and we discuss their behavior under the operation of inserting full twists in the strings of closed braids.
New operations transform braids into P-fibered braids.
problem Transforming braids into P-fibered braids.
method Satellite operations and full twists.
result Any braid can be transformed into a P-fibered braid.
Proves Serre duality in Khovanov-Rozansky homology.
problem Relating top and bottom Hochschild degrees in Khovanov-Rozansky homology.
method Using type A Soergel bimodules and full twist as a Serre functor.
result Categorifies Kálmán's theorem on Hochschild degrees.
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.
Twisted links are obtained from a base link by starting with a n-braid representation, choosing several (m) adjacent strands, and applying one or more twists to the set. Various restrictions may be applied, e.g. the twists may be required to be positive or full twists, or the base braid may be required to have a ce…
Proves a quasi-order on fibre surfaces of positive braid links.
problem Ordering fibre surfaces of positive braid links.
method Analyzes quasipositive surfaces and their inclusions.
result Induced order on fibre surfaces is a well-quasi-order under certain conditions.
New method distinguishes twist equivalence of Seifert surfaces.
problem Distinguishing twist equivalence classes of Seifert surfaces.
method Using bridge spheres and operations on Seifert surfaces.
result Bridge numbers of twist equivalent Seifert surfaces are uniformly bounded.
HZ transform applied to knot polynomials reveals hyperbolic knot structures.
problem Understanding the structure of knot polynomials and their factorisability.
method Applying the Harer-Zagier transform to knot polynomials and character expansions.
result Construction of an infinite family of hyperbolic knots and proof of factorisability in the 3-strand case.
The paper studies hyperbolic geometry of links formed by adding trivial components to knots.
problem Characterizing hyperbolic geometry in links formed by adding trivial components to knots.
method Examining generalized augmented links of knots given by positive braids with at least one full twist.
result Conditions for the hyperbolic geometry of these links are identified.
The paper classifies twisted torus links that are unlinks.
problem Identifying conditions for twisted torus links to be unlinks.
method Characterization of parameter families (p,q,r,s) for unlinking twisted torus links. result Complete characterization of parameter families for unlinking twisted torus links.
Study links using Soergel bimodules and Serre duality.
problem Computing the triply-graded homology of the Whitehead link.
method Representability of partial trace functors and diagrammatics for Hochschild cohomology.
result Computed the triply-graded homology of the Whitehead link.
We use some Lie group theory and Budney's unitarization of the Lawrence-Krammer representation, to prove that for generic parameters of definite form the image of the representation (also on certain types of subgroups) is dense in the unitary group. This implies that, except possibly for closures of full-twist braids, …
Corrected mistakes in knot Floer homology formulas for spliced knot complements.
problem Incorrect assumption about knot Floer homology involution.
method Considered the contribution from a possibly non-trivial involution.
result Corrected splicing formulas in knot Floer homology.
Study proves bridge and braid indices match for twist positive knots.
problem Determining when bridge and braid indices are equal for knots.
method Used knot Floer torsion order to prove for all twist positive knots.
result Bridge and braid indices coincide for all twist positive knots.
Knot homology of Coxeter links identified with line bundles on Hilbert schemes.
problem Identifying knot homology for Coxeter links.
method Using line bundles on a generalized flag Hilbert scheme of points in \(\mathbb{C}^2\).
result Knot homology of Coxeter links corresponds to sections of a specific line bundle.
We conjecture formulae of the colored superpolynomials for a class of twist knots Kp where p denotes the number of full twists. The validity of the formulae is checked by applying differentials and taking special limits. Using the formulae, we compute both the classical and quantum super-A-polynomial for the twist k…
Categorifies a skein relation for links colored by one-column Young diagrams.
problem Categorification of a skein relation for links.
method Homotopy equivalence between two complexes of singular Soergel bimodules.
result Categorification of the colored skein relation.
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.
In this paper we show that the non-alternating torus knots are homologically thick, i.e. that their Khovanov homology occupies at least three diagonals. Furthermore, we show that we can reduce the number of full twists of the torus knot without changing certain part of its homology, and consequently, we show that there…
Study categorifies link invariants using Soergel bimodules.
problem Categorification of link invariants.
method Explicit computation of derived traces of Soergel bimodules.
result Derived annular Khovanov-Rozansky link invariant.
For a closed n-braid L with a full positive twist and with k negative crossings, 0\leq k \leq n, we determine the first n-k+1 terms of the Jones polynomial V_L(t). We show that V_L(t) satisfies a braid index constraint, which is a gap of length at least n-k between the first two nonzero coefficients of (1-t^2)V_L(t). F…
New characterization of symplectic surfaces in CP^2 via bridge trisections.
problem Characterize symplectic surfaces in CP^2.
method Bridge trisections and quasipositive factorizations.
result Minimal genus symplectic surfaces are isotopic to surfaces in transverse bridge position.
Paper calculates the ribbonlength of twisted torus knots.
problem Determining the ribbonlength of twisted torus knots.
method Analyzes the ribbonlength of twisted torus knots Tp,q;r,s and provides upper bounds. result Ribbonlength of Tp,q;r,s is bounded by 2(max{p,q,r}+∣s∣r) and 2(p+(∣s∣−1)r) under specific conditions. Knots from a specific band sum have similar homologies but are distinct.
problem Cosmetic crossing conjecture for knots from a specific band sum.
method Analyzing knot homologies (Heegaard, instanton, Khovanov) to verify distinctness.
result Knots from the band sum are distinct despite similar homologies.
These are Lecture Notes of a course given by the author at the French-Spanish School "Tresses in Pau", held in Pau (France) in October 2009. It is basically an introduction to distinct approaches and techniques that can be used to show results in braid groups. Using these techniques we provide several proofs of well kn…
Braids with more twists than strands achieve their braid index.
problem Understanding the braid index of closures of braids with fractional Dehn twist coefficients.
method Characterizing fractional Dehn twist coefficients in terms of the Upsilon function and proving the braid index via homogenization of knot concordance homomorphisms.
result Proves that n-braids with fractional Dehn twist coefficient larger than n−1 realize the braid index of their closure. New method encodes knots using clasp diagrams for easier study of invariants.
problem Encoding and studying knots and links more efficiently.
method Introducing clasp diagrams and equivalence relations on them.
result Explicit formula for Alexander-Conway polynomial and Vassiliev invariants.
Extended symmetric unions extend properties of Alexander polynomials.
problem Properties of Alexander polynomials of symmetric unions.
method Constructing pairs of knots with specific epimorphisms.
result Alexander polynomials of constructed knots exhibit extended properties.
Area-preserving diffeomorphisms of a 2-disc can be regarded as time-1 maps of (non-autonomous) Hamiltonian flows on solid tori, periodic flow-lines of which define braid (conjugacy) classes, up to full twists. We examine the dynamics relative to such braid classes and define a braid Floer homology. This refinement of t…