arXiv research

Defines a new Upsilon torsion function for knot Floer homology.

This paper shows hyperbolic knots can have arbitrarily large torsion in knot Floer homology.

Grid homology confirms the Upsilon invariant in knot theory.

New knots share same Upsilon invariant despite different Alexander polynomials.

New hyperbolic knots with convex Upsilon invariants constructed.

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.

In this paper we construct an infinite family of knots with vanishing Upsilon invariant $Υ$, although their secondary Upsilon invariants $Υ^2$ show that they are linearly independent in the smooth knot concordance group. We also prove a conjecture in a paper by Allen.

We give a formula of the Upsilon invariant of any L-space cable knot $K_{p,q}$ using $p,Υ_K$ and $Υ_{T_{p,q}}$. The integral value of the Upsilon invariant gives a ${\mathbb Q}$-valued knot concordance invariant. We compute the integral values for L-space iterated cable 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, …

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…

New deformations of lattice cohomology help calculate knot invariants.

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…

The Upsilon invariant helps classify fibered knots and their open book decompositions.

The paper extends a knot invariant to graphs and connects it to homology cylinders.

Hom gives an example of a knot with vanishing Upsilon invariant but nonzero epsilon invariant. We build more such knots that are linearly independent in the smooth concordance group.

The knot invariant Upsilon, defined by Ozsvath, Stipsicz, and Szabo, induces a homomorphism from the smooth knot concordance group to the group of piecewise linear functions on the interval [0,2]. Here we define a set of related secondary invariants, each of which assigns to a knot a piecewise linear function on [0,2].…

Two Heegaard Floer knot complexes are called stably equivalent if an acyclic complex can be added to each complex to make them filtered chain homotopy equivalent. Hom showed that if two knots are concordant, then their knot complexes are stably equivalent. Invariants of stable equivalence include the concordance invari…

We use Ozsváth, Stipsicz, and Szabó's Upsilon-invariant to provide bounds on cobordisms between knots that `contain full-twists'. In particular, we recover and generalize a classical consequence of the Morton-Franks-Williams inequality for knots: positive braids that contain a positive full-twist realize the braid inde…

New concordance invariants phi and phi_j are defined and studied.

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.

Using the theory of involutive Heegaard Floer knot theory developed by Hendricks-Manolescu, we define two involutive analogs of the Upsilon knot concordance invariant of Ozsvath-Stipsicz-Szabo. These involutive invariants are piecewise linear functions defined on the interval [0,2]. Each is a concordance invariant and …

We extend the construction of upsilon-type invariants to null-homologous knots in rational homology three-spheres. By considering $m$-fold cyclic branched covers with $m$ a prime power, this extension provides new knot concordance invariants $Υ_m^C (K)$ of knots in $S^3$. We give computations of these invariants for so…

The paper calculates a knot invariant for 3-braid knots.

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…

We construct smooth concordance invariants of knots which take the form of piecewise linear maps from [0,1] to R, one for each n greater than or equal to 2. These invariants arise from sl(n) knot cohomology. We verify some properties which are analogous to those of the invariant Upsilon (which arises from knot Floer ho…

To a region $C$ of the plane satisfying a suitable convexity condition we associate a knot concordance invariant $Υ^C$. For appropriate choices of the domain this construction gives back some known knot Floer concordance invariants like Rasmussen's $h_i$ invariants, and the Ozsv\' ath-Stipsicz-Szab\' o upsilon invarian…

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…

We study the effect of satellite operations on the Upsilon invariant of Ozsvath-Stipsicz-Szabo. We obtain results concerning when a knot and its satellites are independent; for example, we show that the set $\{D_{2^i,1}\}_{i=1}^\infty$ is a basis for an infinite rank summand of the group of smooth concordance classes o…

We characterize the fractional Dehn twist coefficient of a braid in terms of a slope of the homogenization of the Upsilon function, where Upsilon is the function-valued concordance homomorphism defined by Ozsváth, Stipsicz, and Szabó. We use this characterization to prove that $n$-braids with fractional Dehn twist coef…

We use virtual knot theory to detect the non-invertibility of some classical links in $\mathbb{S}^3$. These links appear in the study of virtual covers. Briefly, a virtual cover associates a virtual knot $\upsilon$ 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 $L=J \sqcup K$ with $J$ fibered. These are concordances that restrict to fibered concordances on the first …

Improved regret bounds for linear bandits with heavy-tailed rewards.

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.

We give asymptotically sharp upper bounds for the Khovanov width and the dealternation number of positive braid links, in terms of their crossing number. The same braid-theoretic technique, combined with Ozsváth, Stipsicz, and Szabó's Upsilon invariant, allows us to determine the exact cobordism distance between torus …

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…

We study the asymptotic growth of the eigenvalues of the Laplace-Beltrami operator on singular Riemannian manifolds, where all geometrical invariants appearing in classical spectral asymptotics are unbounded, and the total volume can be infinite. Under suitable assumptions on the curvature blow-up, we show how the sing…

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…

New homomorphism from Khovanov homology gives slice genus bounds.

We show that there exists a $\mathbb{Z}^\infty$-summand in the subgroup of the knot concordance group generated by knots with trivial Alexander polynomial. To this end we use the invariant Upsilon $Υ$ recently introduced by Ozsváth, Stipsicz and Szabó using knot Floer homology. We partially compute $Υ$ of $(n,1)$-cable…

Algebraic knots are known to be iterated torus knots and to admit L-space surgeries. However, Hedden proved that there are iterated torus knots that admit L-space surgeries but are not algebraic. We present an infinite family of such examples, with the additional property that no nontrivial linear combination of knots …

By considering negative surgeries on a knot $K$ in $S^3$, we derive a lower bound to the non-orientable slice genus $γ_4(K)$ in terms of the signature $σ(K)$ and the concordance invariants $V_i(\overline{K})$, which strengthens a previous bound given by Batson, and which coincides with Ozsváth-Stipsicz-Szabó's bound in…

In this paper, we study the behavior of $Υ_K(t)$ under the cabling operation, where $Υ_K(t)$ is the knot concordance invariant defined by Ozsváth, Stipsicz, and Szabó, associated to a knot $K\subset S^3$. The main result is an inequality relating $Υ_K(t)$ and $Υ_{K_{p,q}}(t)$, which generalizes the inequalities of Hedd…

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…

Analytic torsion and Reidemeister torsion help show exponential growth of torsion in cohomology.

Explicitly expresses torsion functions on lens spaces.

The paper compares two torsion invariants in complex vector bundles.