Khovanov Homotopy Calculations using Flow Category Calculus

Abstract

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 flow category simplification techniques previously defined by the authors and Dan Jones. We give a purely combinatorial approach to calculating the second Steenrod square and Bockstein homomorphisms in Khovanov cohomology, and flow categories in general.The new method has been implemented in a computer program by the third author and applied to large classes of knots and links. Several homotopy types not previously witnessed are observed, and more evidence is obtained that Khovanov stable homotopy types do not contain CP 2 as a wedge summand. In fact, we are led by our calculations to formulate an even stronger conjecture in terms of Z/2 summands of the cohomology. 1

A FAST ALGORITHM FOR CALCULATING S-INVARIANTS

“…If we stack crossings as in (3.1) on top of each other, we can do cancellations directly. It is not clear that this improves efficiency, but it can be combined with [11] to make the Sq 2 -refinement below more calculation friendly. Let us define the tangles…”

“…The second Steenrod square requires the stable homotopy type of [12], or at least a 1-flow category as in [11]. At the moment there is no analog of [2] for the stable homotopy type, so we need a global approach to an appropriate 1-flow category.…”

“…We only need to keep track of objects in homological gradings between −3 and 3 and use the cancellation technique of [11] for objects of the same quantum degree. This way we can get much smaller 1-flow categories whose cochain complexes can be used to read off the Khovanov cohomology.…”

Khovanov homotopy type, periodic links and localizations

“…There appeared several algorithms for computing Steenrod squares in Khovanov homology, so the invariants based on Steenrod squares can be effectively computed (see [ 35 , 36 ]). The knotkit package [ 42 ] implements the algorithm of [ 35 ].…”

Categorical lifting of the Jones polynomial: a survey

“…The operation Sq 2 is enough to determine the stable homotopy type for all knots up to 14 crossings and, in fact, some pairs of knots with isomorphic Khovanov homologies are distinguished by their Steenrod squares [159]. By introducing simplification operations for flow categories, one can give computer computations for some more complicated knots, and even by-hand computations for simple knots with nontrivial Sq 2 operations, like the (3,4) torus knot [73,112].…”