[Paper Review] An sl_n stable homotopy type for matched diagrams

Dan Jones, Andrew Lobb+1 more|arXiv (Cornell University)|Jun 25, 2015

Geometric and Algebraic Topology5 references3 citations

TL;DR

This paper constructs an $ɑ\mathfrak{sl}_n$ stable homotopy type for matched diagrams—knot diagrams composed of open 2-braids with oppositely oriented strands and an even number of crossings—generalizing the Lipshitz-Sarkar stable homotopy type to higher $n$. For $n=2$, it recovers the original Lipshitz-Sarkar type; for $n\geq3$, its cohomology recovers $\mathfrak{sl}_n$ Khovanov-Rozansky cohomology, with explicit computations revealing new stable homotopy types, including $\mathbb{C}\mathbf{P}^2$ in $\mathfrak{sl}_3$ and previously undetected summands in $\mathfrak{sl}_4$.

ABSTRACT

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 stable homotopy type and use it to make new computations. Similarly, there exists a simplified Khovanov-Rozansky sl_n complex for open 2-braids with oppositely oriented strands and an even number of crossings. Diagrams made by gluing tangles of this type are called matched diagrams, and knots admitting matched diagrams are called bipartite knots. To a pair consisting of a matched diagram and a choice of integer n >= 2, we associate a stable homotopy type. In the case n = 2 this agrees with the Lipshitz-Sarkar stable homotopy type of the underlying knot. In the case n >= 3 the cohomology of the stable homotopy type agrees with the sl_n Khovanov-Rozansky cohomology of the underlying knot. We make some consistency checks of this sl_n stable homotopy type and show that it exhibits interesting behaviour. For example we find a CP^2 in the sl_3 type for some diagram, and show that the sl_4 type can be interesting for a diagram for which the Lipshitz-Sarkar type is a wedge of Moore spaces.

Motivation & Objective

• Develop a stable homotopy type for $\mathfrak{sl}_n$ Khovanov-Rozansky cohomology that generalizes the Lipshitz-Sarkar construction beyond $n=2$.
• Address the computational challenge of higher $\mathfrak{sl}_n$ invariants by restricting to matched diagrams, which admit simplified complexes.
• Construct a framed flow category from glued tangle diagrams to realize the stable homotopy type via the Cohen-Jones-Segal machinery.
• Verify consistency of the construction by checking agreement with known invariants, including the second Steenrod square and integral cohomology.
• Enable new computations of stable homotopy types for knots like $T_{4,7}$ and $P(2,-3,5)$, revealing previously undetected summands.

Proposed method

• Define matched diagrams as link diagrams composed of open 2-braids with opposite strand orientations and even crossings, admitting a simplified $\mathfrak{sl}_n$ complex for $n\geq2$.
• Construct a framed flow category from the tangle decomposition using the 'sock flow category' and its covering, generalizing the Lipshitz-Sarkar approach.
• Apply the Cohen-Jones-Segal construction to the framed flow category to produce a stable homotopy type whose cohomology recovers $\mathfrak{sl}_n$ Khovanov-Rozansky cohomology.
• Use obstruction theory and frame assignments to ensure the flow category is well-framed and compatible with the stable homotopy type construction.
• Compute the second Steenrod square on the stable homotopy type to detect non-trivial topology, particularly in quantum degrees with non-vanishing $\mathrm{Sq}^2$.
• Combine Steenrod algebra actions and integral cohomology data with classification theorems (e.g., Baues-Hennes) to identify indecomposable summands in the stable homotopy type.

Experimental results

Research questions

• RQ1 Can a stable homotopy type be constructed for $\mathfrak{sl}_n$ Khovanov-Rozansky cohomology that generalizes the Lipshitz-Sarkar construction beyond $n=2$?
• RQ2 How does the stable homotopy type behave for matched diagrams, and does it recover the correct $\mathfrak{sl}_n$ cohomology in all cases?
• RQ3 What new topological features emerge in the $\mathfrak{sl}_n$ stable homotopy type for $n\geq3$, particularly in comparison to the $n=2$ case?
• RQ4 Can the second Steenrod square and integral cohomology data be used to detect previously unknown stable homotopy summands in $\mathfrak{sl}_n$ types?
• RQ5 Are there knots for which the $\mathfrak{sl}_n$ stable homotopy type is non-wedge-of-Moore-spaces even when the $n=2$ type is, indicating richer topology?

Key findings

• For $n=2$, the constructed stable homotopy type agrees with the Lipshitz-Sarkar stable homotopy type, confirming consistency with the known case.
• In the $\mathfrak{sl}_3$ case, the stable homotopy type for the $(3,4)$-torus knot contains a $\mathbb{C}\mathbf{P}^2$ summand, indicating non-trivial topology.
• For the $\mathfrak{sl}_4$ type of the $(3,4)$-torus knot, the stable homotopy type is not a wedge of Moore spaces, with integral cohomology showing $H_{-2}(\mathcal{X}^4_{-1}(D)) = \mathbb{Z} \oplus \mathbb{Z}/4$, requiring further Bockstein computation.
• In the $\mathfrak{sl}_4$ type of $T_{4,7}$, a new summand $X(\eta 2, 5)$ appears in quantum degree $q=29$, previously undetected in the Lipshitz-Sarkar type.
• In the $\mathfrak{sl}_4$ type of $T_{4,5}$, the stable homotopy type in quantum degree $q=25$ is identified as $X(\eta 4, 7)$, a previously undetected summand not present in the $n=2$ case.
• Several $\mathfrak{sl}_n$ stable homotopy types for $T_{4,7}$ and $T_{4,5}$ exhibit non-trivial second Steenrod squares in multiple quantum degrees, with summands like $X(_2\eta, 2)$ and $X(\eta 2, 5)$, confirming non-trivial topology beyond Moore spaces.