Abstract
It is well known that generic perturbations of the complex Frobenius algebra used to define Khovanov cohomology each give rise to Rasmussen's 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 and to new sliceness obstructions that are not equivalent to concordance homomorphisms.
Highlights
We demonstrate that sn for n 3 does not arise generically, and that varying the chosen perturbation gives rise both to new concordance homomorphisms and to new sliceness obstructions that are not equivalent to concordance homomorphisms
There is a cohomology theory associated to each degree n monic polynomial ∂w ∈ C[x] which we write as
We refer to ∂w as the potential of the cohomology theory
Summary
The pretzel knot P (2, −3, 5) appears in the knot table as 10125, and we shall refer to this knot as P for the remainder of this subsection (Figure 1 and 2). We are looking for spectral sequences starting from E1-pages the reduced cohomologies of Figures 1 and 2, and which have as their final pages 1-dimensional cohomologies supported in cohomological degree 0. Look for spectral sequences from this E1 page in which all nontrivial differentials decrease the quantum grading by multiples of 2(n − 1) = 8, and in which the final page is again of dimension n = 5 supported in cohomological degree 0.
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