New family of knots with L-space surgeries but not algebraic.
problem Characterizing knots with L-space surgeries.
method Infinite family construction and new semigroup invariant.
result No nontrivial linear combination of knots in the family is concordant to an algebraic knot.
New knot invariants from Floer theory help bound knot three-genus.
problem Bounding the three-genus of knots.
method Involutive Heegaard Floer knot theory.
result Two new concordance invariants defined.
Ozsvath-Stipsicz-Szabo recently defined a one-parameter family, upsilon of K at t, of concordance invariants associated to the knot Floer complex. We compare their invariant to the {-1, 0, 1}-valued concordance invariant epsilon, which is also associated to the knot Floer complex. In particular, we give an example of a…
We study the four-genus of linear combinations of torus knots: aT(p,q) # -bT(p',q'). Fixing positive p, q, p', and q', our focus is on the behavior of the four-genus as a function of positive a and b. Three types of examples are presented: in the first, for all a and b the four-genus is completely determined by the Tri…
The Upsilon invariant helps classify fibered knots and their open book decompositions.
problem Classifying fibered knots and their open book decompositions.
method Using the Ozsváth-Stipsicz-Szabó concordance invariant Upsilon.
result Fibered knots satisfying a specific condition are either unique in their smooth concordance classes or provide counterexamples to the Slice-Ribbon Conjecture.
Study satellite operations on Upsilon invariant of knots.
problem Effect of satellite operations on Upsilon invariant of knots.
method Analyzes satellite operations on Upsilon invariant of Ozsvath-Stipsicz-Szabo.
result Shows independence of knots and their satellites in certain cases.
Study improves bounds on non-orientable slice genus using knot signatures and concordance invariants.
problem Improving bounds on the non-orientable slice genus of knots.
method Negative surgeries on knots, lower bound derivation using signature and concordance invariants.
result Superadditivity of bounds on stable non-orientable genus, sometimes better than bounds on γ4(K). New hyperbolic knots with convex Upsilon invariants constructed.
problem Constructing knots with convex Upsilon invariants.
method Combinatorial method for (1,1)-knots and connected sum operation. result Infinitely many mutually non-concordant hyperbolic knots with convex Upsilon invariants.
New knots share same Upsilon invariant despite different Alexander polynomials.
problem Identifying concordant knots via Upsilon invariant.
method Examined hyperbolic L-space knots and their Upsilon invariants.
result Infinitely many pairs of hyperbolic L-space knots with distinct Alexander polynomials share the same Upsilon invariant.
Given an L-space knot we show that its Upsilon function is the Legendre transform of a counting function equivalent to the d-invariants of its large surgeries. The unknotting obstruction obtained for the Upsilon function is, in the case of L-space knots, contained in the d-invariants of large surgeries. Generalizations…
Defines a new Upsilon torsion function for knot Floer homology.
problem Obtaining constraints on knot cobordisms.
method Defines a one-parameter family of Heegaard Floer torsion invariants.
result Provides new obstructions related to the Gordian distance between knots.
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, …
Paper finds knots with vanishing Upsilon but non-trivial secondary Upsilon.
problem Understanding the Upsilon invariant and its secondary version.
method Constructing infinite families of knots and proving conjectures.
result Secondary Upsilon invariant can be non-trivial even if Upsilon is zero.
Grid homology confirms the Upsilon invariant in knot theory.
problem Verifying the equivalence of Upsilon invariants in knot theory.
method Reconstructed Upsilon invariant using grid homology and proved equivalence.
result Upsilon invariants in knot Floer and grid homology are equivalent.
New knot invariants bound genus and concordance genus.
problem Bounding knot genus and concordance genus.
method Defining secondary Upsilon invariants as piecewise linear functions.
result Secondary invariants detect knots not detected by Upsilon.
New invariant defined for unoriented knots, proving no factorization through topological concordance.
problem Defining and proving properties of unoriented slice-torus invariants.
method Introducing and proving properties of unoriented slice-torus invariants.
result Unoriented slice-torus invariants do not factor through the topological concordance group.
Formula for Upsilon invariant of L-space cable knots derived.
problem Calculating the Upsilon invariant for L-space cable knots.
method Using p,ΥK and ΥTp,q to derive a formula. result Integral values of the Upsilon invariant as a knot concordance invariant.
New knots found with zero Upsilon but nonzero epsilon.
problem Finding knots with specific concordance invariants.
method Constructing new knots with linear independence in the smooth concordance group.
result Found knots with vanishing Upsilon but nonzero epsilon.
We show that the location of the first singularity of the Upsilon function of an algebraic knot is determined by the first term of its Puiseux characteristic sequence. In many cases this gives better bounds than the tau invariant on the genus of a cobordism between algebraic knots.
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. The upsilon distribution, the sum of independent chi random variates and a normal, is introduced. As a special case, the upsilon distribution includes Lecoutre's lambda-prime distribution. The upsilon distribution finds application in Frequentist inference on the Sharpe ratio, including hypothesis tests on independent …
New family of knots with epsilon invariant nonzero despite Upsilon and phi being zero.
problem Comparing smooth concordance invariants.
method Building an infinite family of knots.
result Found knots with epsilon invariant nonzero but Upsilon and phi zero.
New Upsilon invariants rule out stable equivalence of knot complexes.
problem Stable equivalence of knot complexes and its invariants.
method Secondary Upsilon invariants defined by Kim and Livingston.
result Relations between Upsilon invariants do not extend to stable equivalence.
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 deformations of lattice cohomology help calculate knot invariants.
problem Calculating knot invariants using lattice cohomology.
method Using holomorphic triangles counting and lattice cohomology.
result Combinatorial formulae for the upsilon invariant are derived.
The paper extends a knot invariant to graphs and connects it to homology cylinders.
problem Understanding the structure of homology cobordism groups.
method Using tangle Floer homology, the authors define a new invariant for embedded graphs and prove a concatenation formula.
result The new invariant induces a homomorphism on the homology cobordism group of homology cylinders.
This paper shows hyperbolic knots can have arbitrarily large torsion in knot Floer homology.
problem Understanding the torsion order in knot Floer homology for hyperbolic knots.
method Unified approach using Upsilon torsion function.
result Arbitrarily large torsion orders realized by hyperbolic knots, most of which are twisted torus knots.
New knot concordance invariants from cyclic covers of prime power.
problem Constructing new knot concordance invariants.
method Considering m-fold cyclic branched covers with m a prime power.
result Computations of new invariants for some families of knots.
New concordance invariants phi and phi_j are defined and studied.
problem Understanding the relationships between different concordance invariants.
method Defined and analyzed new invariants phi and phi_j, and provided recursive formulas.
result Found infinitely many knots with specific combinations of zero and nonzero phi invariant.
We describe an error in the proof of a key proposition, which was necessary for the proof of the main result. Alternate proofs of the main result are given by Ozsvath-Stipsicz-Szabo and Dai-Hom-Stoffregen-Truong.
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. Upper bounds for Khovanov width and dealternation number derived for positive braids.
problem Determining bounds for Khovanov width and dealternation number of positive braid links.
method Braid-theoretic technique combined with Upsilon invariant.
result Asymptotically sharp upper bounds for Khovanov width and dealternation number in terms of crossing number.
In an earlier paper, we proved that given an asymptotically cylindrical G_2-manifold M with a Calabi-Yau boundary X, the moduli space of coassociative deformations of an asymptotically cylindrical coassociative 4-fold C in M with a fixed special Lagrangian boundary L in X is a smooth manifold of dimension dim(V_+), whe…
Researchers study eigenvalues on singular Riemannian manifolds, showing how curvature affects Weyl's law.
problem Analyzing eigenvalues of Laplace-Beltrami operator on singular Riemannian manifolds with unbounded geometrical invariants.
method Developed a new quantitative estimate for the remainder of the heat trace and Weyl's function on Riemannian manifolds.
result Constructed singular Riemannian metrics with prescribed non-classical Weyl's law for various slowly varying functions.
Combinatorial method computes Legendrian knot invariant.
problem Computing the Heegaard Floer contact invariant for Legendrian knots.
method Combining Plamenevskaya's combinatorial description with Heegaard Floer theory.
result Hat version of LOSS invariant can be computed combinatorially.
New infinite-rank summand found in knot concordance group.
problem Existence of knots with trivial Alexander polynomial and infinite-rank summands.
method Utilized knot Floer homology and the Upsilon invariant.
result Existence of a Z^∞-summand in knot concordance group with trivial Alexander polynomial.
New knot concordance invariants from sl(n) cohomology.
problem Developing new invariants to distinguish knots.
method Construct piecewise linear maps from [0,1] to R, derived from sl(n) knot cohomology.
result Verify properties analogous to Upsilon invariant and define new concordance invariants.
Algorithm computes knot invariants for surgeries on prime knots.
problem Computing specific knot invariants for Dehn surgeries.
method Grid homology theory for explicit computation.
result Computed invariants for all prime knots up to 11 crossings.
New knot concordance invariants derived from regions in the plane.
problem Knot concordance and distinguishing knots from thin or algebraic ones.
method Associate invariants to regions in the plane, compute for specific knots, and use to obstruct concordances.
result Compute and use new invariants to obstruct concordances to specific types of knots.
We use virtual knot theory to detect the non-invertibility of some classical links in S3. These links appear in the study of virtual covers. Briefly, a virtual cover associates a virtual knot υ to a knot K in a 3-manifold N, under certain hypotheses on K and N. Virtual covers of links in …
A fibered concordance of knots, introduced by Harer, is a concordance between fibered knots that is well-behaved with respect to the fibrations. We consider semi-fibered concordance of two component ordered links L=J⊔K with J fibered. These are concordances that restrict to fibered concordances on the first …
Improved regret bounds for linear bandits with heavy-tailed rewards.
problem Stochastic linear bandits with heavy-tailed rewards.
method Elimination-based algorithm guided by experimental design.
result Regret bound of \(\tilde{\mathcal{O}}(d^\frac{1+3ε}{2(1+ε)} T^\frac{1}{1+ε})\) for \(ε\in (0,1)\).
We define a filtration of the smooth concordance group based on the genus of representative knots. We use the Heegaard Floer epsilon and Upsilon invariants to prove the quotient groups with respect to this filtration are infinitely generated. Results are applied to three infinite families of topologically slice knots.
Study automorphisms and subgroups of exceptional Lie group E8.
problem Classify automorphisms and subgroups of the exceptional Lie group E8.
method Explicitly defined automorphisms and determined subgroups of E8.
result Global realizations of three Riemannian 4-symmetric spaces.
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…
The paper derives and proves the Helgason Fourier transform for vector bundle-valued differential forms on homogeneous spaces.
problem Deriving the Helgason Fourier transform for vector bundle-valued differential forms on homogeneous spaces.
method Employing the perspective of the functional equation satisfied by the classical Fourier transform, the paper derives the Helgason Fourier transform map and proves its properties.
result The Fourier transform is explicitly given and proven to be a map from vector bundle-valued differential forms to another vector bundle-valued differential form on the product space.
In this article, we derive concentration inequalities for the cross-validation estimate of the generalization error for subagged estimators, both for classification and regressor. General loss functions and class of predictors with both finite and infinite VC-dimension are considered. We slightly generalize the formali…
Let xi be a smooth oriented vector bundle, with n-dimensional fibre, over a smooth manifold M. Denote by xi-hat the fibrewise one-point compactification of xi. The main purpose of this paper is to define geometrically a canonical element Upsilon(xi) in H^n(xi-hat,Q) (H^n(xi-hat,Z) tensor 1/2, to be more precise). The e…