Khovanov homology detects the Hopf links

Abstract

We prove that any link in S 3 whose Khovanov homology is the same as that of a Hopf link must be isotopic to that Hopf link. This holds for both reduced and unreduced Khovanov homology, and with coefficients in either Z or Z/2Z. Khovanov homology [Kho00] associates to each link L ⊂ S 3 a bigraded group Kh * , * (L), whose graded Euler characteristic recovers the Jones polynomial V L (q), as well as a reduced variant Kh * , * red (L) [Kho03]. It is known to detect the unknot [KM11], the n-component unlink for all n [BS15], and the trefoils [BS18]. In this note we prove the same for the Hopf links H ± , which are oriented so that the two components have linking number ±1. Thenwhere Z (h,q) is a copy of Z in bigrading (h, q). The reduced Khovanov homology of a link depends on a choice of distinguished component, which we generally suppress from the notation, but we have Kh red (H + ; Z) ∼ = Z (0,1) ⊕ Z (2,5) and Kh red (H − ; Z) ∼ = Z (0,−1) ⊕ Z (−2,−5) regardless of this choice.Theorem 1. Let L be a link in S 3 such that either Kh(L) ∼ = Kh(H ± ) or Kh red (L) ∼ = Kh red (H ± ) as bigraded groups, with coefficients in either Z or Z/2Z. Then L = H ± .

Khovanov homology detects the figure‐eight knot

“…In summary, we have shown that K is either the figure-eight knot or, up to mirroring, a genus-2, fibered, strongly quasi-positive knot with the same knot Floer homology as T (2,5). It remains to show that d = 0 in the first case and d = 2 in the second.…”

“…In 2010, Kronheimer and Mrowka used instanton gauge theory to prove that Khovanov homology detects the unknot [13]. In 2018, Baldwin and Sivek combined gauge theory with ideas from contact topology to prove that Khovanov homology detects the trefoils [1]. Until now, these were the only knots known to be detected by Khovanov homology, though there have been several additional results (most very recent) regarding detection of links whose components are unknots and trefoils; see Hedden–Ni [8], Batson–Seed [4], Baldwin–Sivek–Xie [2], Xie–Zhang [24, 25], Lipshitz–Sarkar [15], Martin [17] and Li–Xie–Zhang [14].…”

“…(2) if d = 0, then, up to mirroring, d = 2 and K is a genus-2, fibered, strongly quasi-positive knot whose bigraded knot Floer homology over Q is isomorphic to that of the torus knot T (2,5).…”

Khovanov homology detects $T(2,6)$

“…Kronheimer and Mrowka showed that Khovanov homology detects the unknot [14]. Khovanov homology is also known to detect the unlink [9] [6], the Hopf link [5], the trefoil [4], the connected sum of two Hopf links [22], the torus link T (2, 4) [22], and split links [16].…”

Two detection results of Khovanov homology on links

“…Since then, many other detection results of Khovanov homology have been obtained. It is now known that Khovanov homology detects the unlink [BS15,HN13], the trefoil [BS18], the Hopf link [BSX19], the forest of unknots [XZ19], the splitting of links [LS19], and the torus link T (2, 6) [Mar20].…”