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A calculus for flow categories (2022)
Journal Article
Lobb, A., Orson, P., & Schuetz, D. (2022). A calculus for flow categories. Advances in Mathematics, 409(Part B), Article 108665. https://doi.org/10.1016/j.aim.2022.108665

We describe a calculus of moves for modifying a framed flow category without changing the associated stable homotopy type. We use this calculus to show that if two framed flow categories give rise to the same stable homotopy type of homological width... Read More about A calculus for flow categories.

A scanning algorithm for odd Khovanov homology (2022)
Journal Article
Schuetz, D. (2022). A scanning algorithm for odd Khovanov homology. Algebraic & geometric topology, 22, 1287-1324. https://doi.org/10.2140/agt.2022.22.1287

We adapt Bar-Natan’s scanning algorithm for fast computations in (even) Khovanov homology to odd Khovanov homology. We use a mapping cone construction instead of a tensor product, which allows us to deal efficiently with the more complicated sign ass... Read More about A scanning algorithm for odd Khovanov homology.

A fast Algorithm for calculating S-Invariants (2020)
Journal Article
Schuetz, D. (2021). A fast Algorithm for calculating S-Invariants. Glasgow Mathematical Journal, 63(2), 378-399. https://doi.org/10.1017/s0017089520000257

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 Rasmussen s-invariants of knots. By disregarding ge... Read More about A fast Algorithm for calculating S-Invariants.

Torsion calculations in Khovanov cohomology (2020)
Journal Article
Schuetz, D. (2020). Torsion calculations in Khovanov cohomology. Journal of Knot Theory and Its Ramifications, 29(8), Article 2071001. https://doi.org/10.1142/s0218216520710017

We obtain information on torsion in Khovanov cohomology by performing calculations directly over Z/pkZ for p prime and k≥2. In particular, we get that the torus knots T(9,10) and T(9,11) contain torsion of order 9 and 27 in their Khovanov cohomology. Read More about Torsion calculations in Khovanov cohomology.

An sl(n) stable homotopy type for matched diagrams (2019)
Journal Article
Jones, D., Lobb, A., & Schuetz, D. (2019). An sl(n) stable homotopy type for matched diagrams. Advances in Mathematics, 356, Article 106816. https://doi.org/10.1016/j.aim.2019.106816

There exists a simplified Bar-Natan Khovanov complex for open 2-braids. The Khovanov cohomology of a knot diagram made by gluing tangles of this type is therefore often amenable to calculation. We lift this idea to the level of the Lipshitz-Sarkar sta... Read More about An sl(n) stable homotopy type for matched diagrams.

Khovanov homotopy calculations using flow category calculus (2019)
Journal Article
Lobb, A., Orson, P., & Schuetz, D. (2020). Khovanov homotopy calculations using flow category calculus. Experimental Mathematics, 29(4), 475-500. https://doi.org/10.1080/10586458.2018.1482805

The Lipshitz–Sarkar stable homotopy link invariant defines Steenrod squares on the Khovanov cohomology of a link. Lipshitz–Sarkar constructed an algorithm for computing the first two Steenrod squares. We develop a new algorithm which implements the f... Read More about Khovanov homotopy calculations using flow category calculus.

Framed cobordism and flow category moves (2018)
Journal Article
Lobb, A., Orson, P., & Schuetz, D. (2018). Framed cobordism and flow category moves. Algebraic & geometric topology, 18, 2821-2858. https://doi.org/10.2140/agt.2018.18.2821

Framed flow categories were introduced by Cohen, Jones and Segal as a way of encoding the flow data associated to a Floer functional. A framed flow category gives rise to a CW complex with one cell for each object of the category. The idea is that th... Read More about Framed cobordism and flow category moves.

A Khovanov stable homotopy type for colored links (2017)
Journal Article
Lobb, A., Orson, P., & Schuetz, D. (2017). A Khovanov stable homotopy type for colored links. Algebraic & geometric topology, 17(2), 1261-1281. https://doi.org/10.2140/agt.2017.17.1261

We extend Lipshitz-Sarkar's definition of a stable homotopy type associated to a link L whose cohomology recovers the Khovanov cohomology of L. Given an assignment c (called a coloring) of positive integer to each component of a link L, we define a s... Read More about A Khovanov stable homotopy type for colored links.

Intersection homology of linkage spaces in odd dimensional Euclidean space (2016)
Journal Article
Schuetz, D. (2016). Intersection homology of linkage spaces in odd dimensional Euclidean space. Algebraic & geometric topology, 16(1), 483-508. https://doi.org/10.2140/agt.2016.16.483

We consider the moduli spaces Md(ℓ)ℳd(ℓ) of a closed linkage with nn links and prescribed lengths ℓ∈Rnℓ∈ℝn in dd–dimensional Euclidean space. For d>3d>3 these spaces are no longer manifolds generically, but they have the structure of a pseudomanifold... Read More about Intersection homology of linkage spaces in odd dimensional Euclidean space.

Intersection homology of linkage spaces (2015)
Journal Article
Schuetz, D. (2016). Intersection homology of linkage spaces. Journal of Topology and Analysis, 08(01), 25-58. https://doi.org/10.1142/s1793525316500023

We consider the moduli spaces ℳd(ℓ) of a closed linkage with n links and prescribed lengths ℓ ∈ ℝn in d-dimensional Euclidean space. For d > 3 these spaces are no longer manifolds generically, but they have the structure of a pseudomanifold. We use i... Read More about Intersection homology of linkage spaces.

On singular foliations on the solid torus (2013)
Journal Article
Arraut, J., Martins, L., & Schuetz, D. (2013). On singular foliations on the solid torus. Topology and its Applications, 160(13), 1659-1674. https://doi.org/10.1016/j.topol.2013.06.012

We study smooth foliations on the solid torus S1×D2S1×D2 having S1×{0}S1×{0} and S1×∂D2S1×∂D2 as the only compact leaves and S1×{0}S1×{0} as singular set. We show that all other leaves can only be cylinders or planes, and give necessary conditions fo... Read More about On singular foliations on the solid torus.

Homology of moduli spaces of linkages in high-dimensional Euclidean space (2013)
Journal Article
Schuetz, D. (2013). Homology of moduli spaces of linkages in high-dimensional Euclidean space. Algebraic & geometric topology, 13(2), 1183-1224. https://doi.org/10.2140/agt.2013.13.1183

We study the topology of moduli spaces of closed linkages in ℝd depending on a length vector ℓ ∈ ℝn. In particular, we use equivariant Morse theory to obtain information on the homology groups of these spaces, which works best for odd d. In the case... Read More about Homology of moduli spaces of linkages in high-dimensional Euclidean space.