In the Khovanov homology of links, presence of -torsion is a very common phenomenon. Finite number of examples of knots with -torsion for n > 2 were also known, none for n > 8. In this article, we present several infinite families of links whose Khovanov homology contains -torsion for 2 < n < 9 and -torsion for s < 24. We introduce 4-braid links with -torsion which are counterexamples to parts of the PS braid conjecture. We also provide an infinite family of knots with -torsion in reduced Khovanov homology and -torsion in odd Khovanov homology.

We prove that the potential extreme Khovanov cohomology of a link is the cohomology of the independence simplicial complex of its Lando graph. We also provide a family of knots having as many non-trivial extreme Khovanov cohomology modules as desired, that is, examples of H-thick knots that are as far from being H-thin as desired.

Given an m-periodic link $$L\subset S^3$$L ⊂ S 3 , we show that the Khovanov spectrum $$\mathcal {X}_L$$X L constructed by Lipshitz and Sarkar admits a group action. We relate the Borel cohomology of $$\mathcal {X}_L$$X L to the equivariant Khovanov homology of L constructed by the second author. The action of Steenrod algebra on the cohomology of $$\mathcal {X}_L$$X L gives an extra structure of the periodic link. Another consequence of our construction is an alternative proof of the localization formula for Khovanov homology, obtained first by Stoffregen and Zhang. By applying the Dwyer–Wilkerson theorem we express Khovanov homology of the quotient link in terms of equivariant Khovanov homology of the original link.

Braid combing is a procedure defined by Emil Artin to solve the word problem in braid groups. It is well known to have exponential complexity. In this paper, we use the theory of straight line programs to give a polynomial algorithm which performs braid combing. This procedure can be applied to braids on surfaces, providing the first algorithm (to our knowledge) which solves the word problem for braid groups on surfaces with boundary in polynomial time and space. In the case of surfaces without boundary, braid combing needs to use a section from the fundamental group of the surface to the braid group. Such a section was shown to exist by Gonçalves and Guaschi, who also gave a geometric description. We propose an algebraically simpler section, which we describe explicitly in terms of generators of the braid group, and we show why the above procedure to comb braids in polynomial time does not work in this case.