
Khovanov Homology of Positive Knots
ICERM - May 2023

by Marc Kegel and Marithania Silvero
(based on joint work with Naageswaran Manikandan and Leo Mousseau [KMMS23])
Knot theory is the branch of mathematics studying mathematical knots. One can create a physical model of a knot by taking a rope, tangling it in space, and then gluing its ends together. Some examples of knots are shown in Figures 1 and 3.

Often knots are studied via so-called knot diagrams, generic projections to 2-planes with stored over and under information at each crossing, as already done silently in Figure 1.
Diagrams (and therefore knots) can be oriented by choosing one of the two possible directions when traveling through it. Each crossing in an oriented diagram can be classified into positive or negative, according to the convention shown in Figure 2. An interesting subclass of knots is given by so-called positive knots. A knot is positive if it admits a diagram where all crossings are positive.


In our project, which can be found at [KMMS23], we have developed a different approach to obstruct positivity of knots. For any knot, Khovanov has developed a computable knot invariant called Khovanov homology, which is given by a doubly graded sequence of abelian groups. We show that some of these Khovanov homology groups take special values for positive knots, extending and generalizing earlier work from [Kho03, PS14, Sto10]. Then we can for example compute the Khovanov homology of o9_34097 and observe that it does not fulfill these properties. Thus we have shown that o9_34097 does not admit any positive diagram.
The same approach works to show that other knots are not positive for which it was previously unknown. Although our obstruction is known to be not perfect, it seems to be strong. Among the 2977 prime knots with at most 12 crossings, exactly 246 are positive and the Khovanov homology obstructs all the other 2731 knots from being positive.
Positive knots are closely related to braids, the topic of the semester program at ICERM (February - May 2022) where this project started [CEE+22].
References
[CEE+22] M. Culler, B. Elias, J. Etnyre, B. Farb, J. González-Meneses, M. Hedden, K. Kawamuro, A. Licata, and J. Licata, Semester program on Braids, Institute for Computational and Experimental Research in Mathematics (ICERM), Providence, Rhode Island (2022).
[Dun20] N. M. Dunfield, A census of exceptional Dehn fillings, Characters in low-dimensional topology, Contemp. Math., vol. 760, Amer. Math. Soc., [Providence], RI, [2020] © 2020, pp. 143–155. MR 4193924
[Kho03] M. Khovanov, Patterns in knot cohomology I, Experiment. Math. 12 (2003), 365–374. MR 2034399
[KMMS23] M. Kegel, N. Manikandan, L. Mousseau, and M. Silvero, Khovanov homology of positive links and of L-space knots, 2023, arXiv:2304.13613.
[PS14] J. H. Przytycki and R. Sazdanović, Torsion in Khovanov homology of semi-adequate links, Fund. Math. 225 (2014), 277–304. MR 3205574
[Sto10] M. Stošić, On conjectures about positive braid knots and almost alternating torus knots, J. Knot Theory Ramifications 19 (2010), 1471–1486. MR 2746199
[Swe] F. Swenton, KLO (Knot-Like Objects), http://KLO-Software.net.
Humboldt Universität zu Berlin, Germany
Email address: kegemarc@hu-berlin.de, kegelmarc87@gmail.com
Universidad de Sevilla, Spain
Email address: marithania@us.es