arXiv research

We define a third grading on Khovanov homology, which is an invariant of annular links but changes by $\pm 1$ under stabilization. We illustrate the use of our computer implementation, and give some example calculations.

New link detection results using closures of 3-braids.

In this paper we shall establish that properly embedded constant mean curvature one surfaces in H^3 of finite topology are of finite total curvature and each end is regular. In particular, this implies the horosphere is the only simply connected such example, and the catenoid cousins the only annular examples of this n…

We construct a braid conjugacy class invariant $κ$ by refining Plamenevskaya's transverse element $ψ$ in Khovanov homology via the annular grading. While $κ$ is not an invariant of transverse links, it distinguishes some braids whose closures share the same classical invariants but are not transversely isotopic. Using …

We use a geometric construction to exhibit examples of autonomous Lagrangian systems admitting exactly two homoclinics emanating from a nondegenerate maximum of the potential energy and reaching a regular level of the potential having the same value of the maximum point. Similarly, we show examples of Hamiltonian syste…

The paper explores new translating solitons and their relation to $Δ$-wings.

Uniform bound on geodesic images for surfaces using bicorn curves.

New examples show non-rotational annuli in a ball, solving a uniqueness problem.

Paper constructs a spectral sequence linking annular Khovanov homology to reduced Khovanov homology.

New symplectic annular Khovanov homology connects knot theory to Floer homology.

Study shows links with unbounded annular Khovanov gradings but bounded Floer gradings.

In this paper, we introduce the annular instanton Floer homology which is defined for links in a thickened annulus. It is an analogue of the annular Khovanov homology. A spectral sequence whose second page is the annular Khovanov homology and which converges to the annular instanton Floer homology is constructed. As an…

A triangulation of a compact 3-manifold is annular-efficient if it is 0-efficient and the only normal, incompressible annuli are thin edge-linking. If a compact 3-manifold has an annular-efficient triangulation, then it is irreducible, boundary-irreducible, and an-annular. Conversely, it is shown that for a compact, ir…

Study detects a specific type of link using annular Khovanov homology.

Researchers create annular translators for mean curvature flow.

We introduce an sl(n) homology theory for knots and links in the thickened annulus. To do so, we first give a fresh perspective on sutured annular Khovanov homology, showing that its definition follows naturally from trace decategorifications of enhanced sl(2) foams and categorified quantum gl(m), via classical skew Ho…

We construct a stable homotopy refinement of quantum annular homology, a link homology theory introduced by Beliakova, Putyra and Wehrli. For each $r\geq 2$ we associate to an annular link $L$ a naive $\mathbb{Z}/r\mathbb{Z}$-equivariant spectrum whose cohomology is isomorphic to the quantum annular homology of $L$ as …

We prove that the Khovanov-Lee complex of an oriented link, L, in a thickened annulus, A x I, has the structure of a bifiltered complex whose filtered chain homotopy type is an invariant of the isotopy class of L in A x I. Using ideas of Ozsvath-Stipsicz-Szabo as reinterpreted by Livingston, we use this structure to de…

New method associates annular links to elements of Thompson's group T.

New equivariant version of Khovanov homology for annuli.

We define an annular version of odd Khovanov homology and prove that it carries an action of the Lie superalgebra $\mathfrak{gl}(1|1)$ which is preserved under annular Reidemeister moves.

The paper extends knot theory to annular and toroidal pseudo knots.

Study proves existence and nonexistence for annular surfaces with specific curvature and boundary.

Extends quantum annular homology to infinite sets.

Proves a conjecture for annular links using homology classes.

Lifts an $\mathfrak{sl}_2$ action to annular Khovanov homology's stable refinement.

Define web algebras for annular SL(2) and SL(3) using foam TQFTs.

New stable homotopy refinement of quantum annular Khovanov homology.

The paper extends knot polynomials to annular and toroidal pseudo links.

In this article, we study the ramification of the Gauss map of complete minimal surfaces in R^m on annular ends. This work is a continuation of previous work of Dethloff-Ha. We thus give an improvement of the results on annular ends of complete minimal surfaces of Jin-Ru.

For a link in a thickened annulus $A \times I$, we define a $\mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z}$ filtration on Sarkar-Seed-Szabó's perturbation of the geometric spectral sequence. The filtered chain homotopy type is an invariant of the isotopy class of the annular link. From this, we define a two-dimensiona…

Let $\widetilde{K}$ be a two-periodic braid and let $K$ be its quotient. In this paper we show there is a spectral sequence from the next-to-top winding number grading of the sutured annular Khovanov homology of the closure of $\widetilde{K}$ to the next-to-top winding number grading of the sutured annular Khovanov hom…

The paper extends knotoid theory to annular and toroidal settings.

New structure for quantum algebra representations.

In this article, we study the ramification of the Gauss map of complete minimal surfaces in R^3 and R^4 on annular ends. We obtain results which are similar to the ones obtained by Fujimoto and Ru for (the whole) complete minimal surfaces, thus we show that the restriction of the Gauss map to an annular end of such a c…

In this paper we describe the notion of an annular end of a Riemann surface being of finite type with respect to some harmonic function and prove some theoretical results relating the conformal structure of such an annular end to the level sets of the harmonic function. We then apply these results to understand and cha…

Develops a new spectrum for annular links, recovering a transverse invariant at extreme gradings.

In this article, we study the modified defect relations of the Gauss map of complete minimal surfaces in $\mathbb R^3$ and $\mathbb R^4$ on annular ends. We obtain results which are similar to the ones obtained by Fujimoto~[J. Differential Geometry \textbf{29} (1989), 245-262] for (the whole) complete minimal surfaces…

Short note observes quantum Hochschild homology as a composition of known operations.

This paper establishes that sutured annular Khovanov homology is not invariant for braid closures under axis-preserving mutations. This follows from an explicit relationship between sutured annular Khovanov homology and the classical Burau representation for braid closures.

Let L be a link in a thickened annulus. We show that its sutured annular Khovanov homology carries an action of the exterior current algebra of the Lie algebra sl_2. When L is an m-framed n-cable of a knot K in the three-sphere, its sutured annular Khovanov homology carries a commuting action of the symmetric group S_n…

This paper upgrades Khovanov homology to an L-infinity module structure.

Research categorizes 2-cobordisms and finds non-finitely axiomatizable equational theories.

We describe the universal target of annular Khovanov-Rozansky link homology functors as the homotopy category of a free symmetric monoidal category generated by one object and one endomorphism. This categorifies the ring of symmetric functions and admits categorical analogues of plethystic transformations, which we use…

For an embedded singly periodic minimal surface M with genus bigger than or equal to 4 and annular ends, some weak symmetry hypotheses imply its congruence with one of the Hoffman-Wohlgemuth examples. We give a very geometrical proof of this fact, along which they come out many valuable clues for the understanding of t…

The paper evaluates homology for links in a solid torus with special boundary conditions.

We prove that for a fixed braid index there are only finitely many possible shapes of the annular Rasmussen $d_t$ invariant of braid closures. Applying the same perspective to the knot Floer invariant $Υ_K(t)$, we show that for a fixed concordance genus of $K$ there are only finitely many possibilities for $Υ_K(t)$. Fo…

Efficient triangulations help in understanding 3-manifold boundaries.