Andrew Lobb

A slice genus lower bound from sl(n) Khovanov–Rozansky homology

We show that a generic perturbation of the doubly-graded Khovanov–Rozansky knot homology gives rise to a lower-bound on the slice genus of a knot. We prove a theorem about obtainable presentations of surfaces embedded in 4-space, which we use to simplify significantly our algebraic computations.

A note on Gornik's perturbation of Khovanov–Rozansky homology

We show that the information contained in the associated graded vector space to Gornik’s version of Khovanov‐Rozansky knot homology is equivalent to a single even integer sn.K/. Furthermore we show that sn is a homomorphism from the smooth knot concordance group to the integers. This is in analogy with Rasmussen’s invariant coming from a perturbation of Khovanov homology. 57M25

Computable bounds for Rasmussen’s concordance invariant

Given a diagram D of a knot K, we give easily computable bounds for Rasmussens concordance invariant s(K). The bounds are not independent of the diagram D chosen, but we show that for diagrams satisfying a given condition the bounds are tight. As a corollary we improve on previously known Bennequin-type bounds on the slice genus.

The Kanenobu knots and Khovanov-Rozansky homology

Kanenobu has given infinite families of knots with the same HOMFLYPT polynomials. We show that these knots also have the same and HOMFLYPT homologies, thus giving the first example of an infinite family of knots indistinguishable by these invariants. This is a consequence of a structure theorem about the homologies of knots obtained by twisting up the ribbon of a ribbon knot with one ribbon.

New quantum obstructions to sliceness

It is well-known that generic perturbations of the complex Frobenius algebra used to define Khovanov cohomology each give rise to Rasmussens concordance invariant s. This gives a concordance homomorphism to the integers and a strong lower bound on the smooth slice genus of a knot. Similar behavior has been observed in sl(n) Khovanov-Rozansky cohomology, where a perturbation gives rise to the concordance homomorphisms s_n for each n >= 2, and where we have s_2 = s. We demonstrate that s_n for n >= 3 does not in fact arise generically, and that varying the chosen perturbation gives rise both to new concordance homomorphisms as well as to new sliceness obstructions that are not equivalent to concordance homomorphisms.

The Quantum sl(N) Graph Invariant and a Moduli Space

We associate a moduli problem to a colored trivalent graph; such graphs, when planar, appear in the state-sum description of the quantum sl(N) knot polynomial due to Murakami, Ohtsuki, and Yamada. We discuss how the resulting moduli space can be thought of a representation variety. We show that the Euler characteristic of the moduli space is equal to the quantum sl(N) polynomial of the graph evaluated at unity. Possible extensions of the result are also indicated.

2–strand twisting and knots with identical quantum knot homologies

Given a knot, we ask how its Khovanov and Khovanov‐Rozansky homologies change under the operation of introducing twists in a pair of strands. We obtain long exact sequences in homology and further algebraic structure which is then used to derive topological and computational results. Two of our applications include giving a way to generate arbitrary numbers of knots with isomorphic homologies and finding an infinite number of mutant knot pairs with isomorphic reduced homologies. 57M25

A Khovanov stable homotopy type for colored links

We extend Lipshitz-Sarkars 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 stable homotopy type X_col(L_c) whose cohomology recovers the c-colored Khovanov cohomology of L. This goes via Rozanskys definition of a categorified Jones-Wenzl projector P_n as an infinite torus braid on n strands. We then observe that Cooper-Krushkals explicit definition of P_2 also gives rise to stable homotopy types of colored links (using the restricted palette {1, 2}), and we show that these coincide with X_col. We use this equivalence to compute the stable homotopy type of the (2,1)-colored Hopf link and the 2-colored trefoil. Finally, we discuss the Cooper-Krushkal projector P_3 and make a conjecture of X_col(U_3) for U the unknot.

ON CASSON-TYPE INSTANTON MODULI SPACES OVER NEGATIVE DEFINITE 4-MANIFOLDS

Recently Andrei Teleman considered instanton moduli spaces over negative definite 4-manifolds X with b2(X) ≥ 1. If b2(X) is divisible by four and b1(X) = 1 a gauge-theoretic invariant can be defined; it is a count of flat connections modulo the gauge group. Our first result shows that if such a moduli space is non-empty and the manifold admits a connected sum decomposition X ≅ X1 # X2, then both b2(X1) and b2(X2) are divisible by four; this rules out a previously naturally appearing source of 4-manifolds with non-empty moduli space. We give in some detail a construction of negative definite 4-manifolds which we expect will eventually provide examples of manifolds with non-empty moduli space.

Framed cobordism and flow category moves

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 the Floer invariant should take the form of the stable homotopy type of the resulting complex, recovering the Floer cohomology as its singular cohomology. Such a framed flow category was produced, for example, by Lipshitz and Sarkar from the input of a knot diagram, resulting in a stable homotopy type generalising Khovanov cohomology. We give moves that change a framed flow category without changing the associated stable homotopy type. These are inspired by moves that can be performed in the Morse–Smale case without altering the underlying smooth manifold. We posit that if two framed flow categories represent the same stable homotopy type then a finite sequence of these moves is sufficient to connect the two categories. This is directed towards the goal of reducing the study of framed flow categories to a combinatorial calculus. We provide examples of calculations performed with these moves (related to the Khovanov framed flow category), and prove some general results about the simplification of framed flow categories via these moves.