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Top Papers in Lipshitz-sarkar's construction of khovanov homotopy type

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Khovanov-Lipshitz-Sarkar homotopy type for links in thickened higher genus surfaces

We define the Khovanov-Lipshitz-Sarkar homotopy type and the Steenrod square
for the homotopical Khovanov homology of links in thickened higher genus
surfaces. Our Khovanov-Lipshitz-Sarkar homotopy ty

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Stable homotopy types for the homotopical Khovanovy of links in thickened surfaces indexed by moduli space systems

Khovanov-Lipshitz-Sarkar homotopy type for links in thickened surfaces and those in $S^3$ with new modulis

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Steenrod square for virtual links toward Khovanov-Lipshitz-Sarkar stable homotopy type for virtual links

We define a second Steenrod square for virtual links, which is stronger than
Khovanov homology for virtual links, toward constructing
Khovanov-Lipshitz-Sarkar stable homotopy type for virtual links. T

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Localization in Khovanov homology

We construct equivariant Khovanov spectra for periodic links, using the
Burnside functor construction introduced by Lawson, Lipshitz, and Sarkar. By
identifying the fixed-point sets, we obtain rank in

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Two second Steenrod squares for odd Khovanov homology

Recently, Sarkar-Scaduto-Stoffregen constructed a stable homotopy type for
odd Khovanov homology, hence obtaining an action of the Steenrod algebra on
Khovanov homology with $\mathbb{Z}/2\mathbb{Z}$ c

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A fast algorithm for calculating $s$-invariants

We use the divide-and-conquer and scanning algorithms for calculating
Khovanov cohomology directly on the Lee- or Bar-Natan deformations of the
Khovanov complex to give an alternative way to compute R

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A flexible construction of equivariant Floer homology and applications

Seidel-Smith and Hendricks used equivariant Floer cohomology to define some
spectral sequences from symplectic Khovanov homology and Heegaard Floer
homology. These spectral sequences give rise to Smit

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gl(2) foams and the Khovanov homotopy type

The Blanchet link homology theory is an oriented model of Khovanov homology,
functorial over the integers with respect to link cobordisms. We formulate a
stable homotopy refinement of the Blanchet the

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Non-Trivial Steenrod Squares on Prime, Hyperbolic and Satellite Knots

We show that there are prime knots so that the Steenrod operations of
Lipshitz and Sarkar arXiv:1204.5776 are non trivial on their Khovanov homology.
This answers a question posed by Lipshitz and Sark

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A Bar-Natan homotopy type

Following Lipshitz-Sarkar's construction of Khovanov homotopy type, we
construct for any link diagram $L$ a CW spectrum $\mathcal{X}_{\mathit{BN}}(L)$
whose reduced cellular cochain complex gives the

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Squeezed Knots

Squeezed knots are those knots that appear as slices of genus-minimizing
oriented smooth cobordisms between positive and negative torus knots. We show
that this class of knots is large and discuss how

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On an integral version of the Rasmussen invariant

We define a Rasmussen $s$-invariant over the coefficient ring of the
integers, and show how it is related to the $s$-invariants defined over a
field. A lower bound for the slice genus of a knot arisin

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Proof of Homotopy Type Theory

On the homotopy groups of spheres in homotopy type theory

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The James construction and $π_4(\mathbb{S}^3)$ in homotopy type theory

In the first part of this paper we present a formalization in Agda of the
James construction in homotopy type theory. We include several fragments of
code to show what the Agda code looks like, and we

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Learning knot invariants across dimensions

We use deep neural networks to machine learn correlations between knot
invariants in various dimensions. The three-dimensional invariant of interest
is the Jones polynomial $J(q)$, and the four-dimens

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Homotopy functoriality for Khovanov spectra

We prove that the Khovanov spectra associated to links and tangles are
functorial up to homotopy and sign.

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Stein trisections and homotopy 4-balls

A homotopy 4-ball is a smooth 4-manifold with boundary $S^3$ that is
homotopy-equivalent to the standard $B^4$. The smooth 4-dimensional Schoenflies
problem asks whether every homotopy 4-ball in $S^4$

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Higher representations and cornered Heegaard Floer homology

We develop the 2-representation theory of the odd one-dimensional super Lie
algebra $gl(1|1)^+$ and show it controls the Heegaard-Floer theory of surfaces
of Lipshitz, Ozsv\'ath and Thurston. Our main

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Khovanov homology and the cinquefoil

We prove that Khovanov homology with coefficients in $\mathbb{Z}/2\mathbb{Z}$
detects the $(2,5)$ torus knot. Our proof makes use of a wide range of deep
tools in Floer homology, Khovanov homology, an

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Poset topology of $s$-weak order via SB-labelings

Ceballos and Pons generalized weak order on permutations to a partial order
on certain labeled trees, thereby introducing a new class of lattices called
$s$-weak order. They also generalized the Tamar

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