# Khovanov Homology Invariants

Khovanov homology is a link invariant due to Khovanov [1] that categorifies the Jones polynomial.

Odd Khovanov homology is a related theory due to Ozsváth, Rasmussen and Szabó [2]. Khovanov homology and odd Khovanov homology have the same reduction mod 2, but are different over Q.

Data for all versions of Khovanov homology displayed in this database were computed using the program KnotJob written by Dirk Schütz, available at [3]. The calculations were performed for several variants of Khovanov homology (odd and original, reduced and unreduced) with different coefficient systems (Z, Q, Z/2Z):

KH Unred Z Poly: Unreduced Khovanov homology calculated over Z

KH Red Z Poly: Reduced Khovanov homology calculated over Z

KH Red Q Poly: Reduced Khovanov homology calculated over Q

KH Red Mod2 Poly: Reduced Khovanov homology calculated over Z/2Z

KH Odd Z Poly: Odd Khovanov homology calculated over Z

KH Odd Q Poly: Odd Khovanov homology calculated over Q

KH Odd Mod2 Poly: Odd Khovanov homology calculated over Z2

Example to clarify notation: The KH Unred Z Poly data (in polynomial format) for the trefoil is

t^(0) q^(1) + t^(0) q^(3) + t^(2) q^(5) + t^(3) q^(9) + t^(2) q^(7) T^(2) + t^(3) q^(7) T^(2)

The first four terms correspond with four Z summands; each is rank one as indicated by the coefficient. The homological and quantum gradings are recorded in the exponents of t, q. The last two summands include a capital T indicating `torsion.' The gradings are recorded as the exponents of t, q, whereas the exponent of T records the modulus of the torsion summand.

Formatted as a vector, this data appears as

[[0, 1, 0, 1], [0, 1, 0, 3], [0, 1, 2, 5], [0, 1, 3, 9], [2, 1, 2, 7], [2, 1, 3, 7]]

Each entry of the vector represents a summand, and is itself a four-component vector. The four components of the vector specify [modulus, rank, homological, quantum] for each summand. For example, [0, 1, 0, 1] indicates a Z-summand of rank 1 in bigrading (0, 1) and [2, 1, 2, 7] indicates a Z/2Z-summand of rank 1 in bigrading (2, 7). A term corresponding with a Q-summand of rank 3 in bigrading (0, 2) would be recorded as [1, 3, 0, 2].

## References

[1] Khovanov, M. (2000), "A categorification of the Jones polynomial", Duke Mathematical Journal, 101 (3): 359--426.

[2] Ozsváth, P. and Rasmussen, J. and Szabó, Z. "Odd Khovanov homology." Algebraic & Geometric Topology, 13(3) 1465--1488 2013.

[3] Schütz, D. KnotJob.