---
title: 'Khovanov Local Equivalence Groups: Overview'
url: https://www.emergentmind.com/topics/khovanov-local-equivalence-groups
type: topic
---

# Khovanov Local Equivalence Groups: Overview

Khovanov local equivalence groups are algebraic concordance targets obtained by identifying Khovanov-type chain complexes whenever maps between them become equivalences on a distinguished localized summand. The principal construction combines odd Khovanov homology with the Bar–Natan deformation of even Khovanov homology into a local even–odd (LEO) triple. Its local-equivalence classes form an abelian group receiving a homomorphism from the smooth knot concordance group. Related constructions use only the Bar–Natan deformation, while the universal integral Khovanov complex yields a further concordance homomorphism whose target retains information invisible to all Rasmussen invariants over fields [2312.09114] [2401.08480].

## 1. Algebraic input and LEO triples

For a commutative ring \(R\), the even Khovanov complex \(CKh(K;R)\) is constructed from the cube of resolutions using the Frobenius algebra
\[
R[X]/(X^2),
\]
with
\[
\Delta(1)=1\otimes X+X\otimes 1,\qquad
\Delta(X)=X\otimes X.
\]
Its differential has bidegree \((1,0)\). The odd Khovanov complex \(CKh_o(K;R)\) uses, at a resolution \(K_v\), the exterior algebra
\[
\Lambda(\pi_0K_v)
\]
instead of a tensor product of copies of \(R[X]/(X^2)\). Over \(\mathbb F_2\), odd and even Khovanov complexes are canonically identified:
\[
CKh_o(K;\mathbb F_2)\cong CKh(K;\mathbb F_2).
\]

For a based knot, odd Khovanov theory carries a right action of \(\mathbb Z[X]/(X^2)\), given by wedging with the basepoint circle. A compatible left action is related to the right action by a chain automorphism, so the right action is used in the local-equivalence construction.

The Bar–Natan complex \(CKh_h(K;R)\) is defined over \(R[h]\) using
\[
R[h,X]/(X^2-hX),
\]
with
\[
\Delta(1)=1\otimes X+X\otimes 1-h\otimes1,\qquad
\Delta(X)=X\otimes X.
\]
The variable \(h\) has quantum degree \(-2\). Setting \(h=0\) recovers ordinary even Khovanov theory:
\[
CKh(K;R)=CKh_h(K;R)\otimes_{R[h]}R[h]/(h).
\]

Localization at \(h\) isolates the reducible part of the Bar–Natan complex. In the reduced setting,
\[
h^{-1}CKh_h(K;R)\simeq h^{-1}R,
\]
while in the unreduced setting the localized homology has two free summands corresponding to the two orientations of the knot:
\[
h^{-1}Kh_h(K;R)\cong R[h,h^{-1}]\oplus R[h,h^{-1}].
\]

A LEO triple is a triple
\[
(C,D,f)
\]
consisting of:

1. A finitely generated, bigraded, freely generated cochain complex \(C\) over
   \[
   \mathbb Z[X]/(X^2).
   \]

2. A finitely generated, bigraded, freely generated cochain complex \(D\) over
   \[
   \mathbb Z[X,h]/(X^2-hX).
   \]

3. A bigraded chain homotopy equivalence
   \[
   f:C\otimes_{\mathbb Z[X]/(X^2)}\mathbb F_2[X]/(X^2)
   \longrightarrow
   D\otimes_{\mathbb Z[X,h]/(X^2-hX)}\mathbb F_2,
   \]
   where the target is obtained by setting \(2=h=0\).

The map \(f\) is required to be a homomorphism over \(\mathbb F_2[X]/(X^2)\). In addition,
\[
h^{-1}D
\]
must be homotopy equivalent to a free graded rank-one module over the localized ring, supported in homological degree \(0\) and in odd quantum gradings.

For a knot \(K\), the motivating triple is
\[
LEO(K)=\bigl(CKh_o(K),CKh_h(K),f\bigr),
\]
where \(f\) is the canonical mod-\(2\) identification. The corresponding even-only construction is
\[
LEE(K)=\bigl(CKh(K),CKh_h(K),f\bigr).
\]

A reduced LEO triple instead uses complexes over \(\mathbb Z\) and \(\mathbb Z[h]\), with localized rank-one homology in even quantum gradings. There is also a two-reduced or mod-\(2\) version over \(\mathbb F_2[h]\).

## 2. Local maps and the abelian group structure

Given LEO triples \((C,D,f)\) and \((C',D',f')\), a local map
\[
(\alpha,\beta):(C,D,f)\longrightarrow(C',D',f')
\]
consists of grading-preserving chain maps
\[
\alpha:C\to C',\qquad \beta:D\to D'
\]
that respect the module structures and satisfy two conditions.

First, after inverting \(h\),
\[
h^{-1}\beta:h^{-1}D\longrightarrow h^{-1}D'
\]
must be a chain homotopy equivalence. Second, the square involving \(f\), \(f'\), \(\alpha\), and \(\beta\) must commute up to a module-compatible chain homotopy.

The local map need not be a chain equivalence before localization. Its defining requirement is equivalence on the localized, or reducible, component. Two LEO triples are locally equivalent when local maps exist in both directions.

The tensor product is
\[
(C,D,f)\otimes(C',D',f')
=
\left(
C\otimes_{\mathbb Z[X]/(X^2)}C'\{-1\},
D\otimes_{\mathbb Z[X,h]/(X^2-hX)}D'\{-1\},
f\otimes f'
\right),
\]
where the quantum shift \(\{-1\}\) is the correction associated with connected sum.

The Künneth theorems for odd Khovanov and Bar–Natan complexes imply
\[
LEO(K_1\# K_2)\sim LEO(K_1)\otimes LEO(K_2).
\]
Dualization supplies inverses. For a triple \((C,D,f)\), its dual is formed by graded module duals, with the differential
\[
d(f)=(-1)^{\operatorname{gr}_h(f)+1}f\circ\partial.
\]
Evaluation and coevaluation maps become homotopy equivalences after localization, yielding
\[
(C,D,f)\otimes(C,D,f)^*\sim LEO(U).
\]

Consequently, local-equivalence classes form an abelian group
\[
C_{\mathrm{LEO}},
\]
with tensor product as operation, the unknot triple as identity, and dualization as inverse. The reduced and two-reduced theories give related groups, commonly denoted \(C_{\mathrm{rLEO}}\) and \(C_{\mathrm{rLEO}^o}\). The unreduced group contains pathological triples that do not arise from knots, so it is larger than the image of the concordance group.

For two-reduced triples, the relation
\[
[(C,D,f)]\geq[(C',D',f')]
\]
is defined by the existence of a local map from the second triple to the first. This relation is a translation-invariant total order. In this setting, the order is characterized by the invariant \(s_o\):
\[
[(C,D,f)]\geq0
\quad\Longleftrightarrow\quad
s_o(C,D,f)\geq0.
\]

## 3. Concordance homomorphisms

An oriented knot cobordism \(\Sigma\) from \(K_0\) to \(K_1\) induces maps on even Khovanov, odd Khovanov, and Bar–Natan complexes. On the even complex, for example,
\[
\Sigma_*:CKh^{i,j}(K_0;R)\longrightarrow CKh^{i,j+\chi(\Sigma)}(K_1;R).
\]
For a concordance, \(\chi(\Sigma)=0\), and the Bar–Natan map becomes a homotopy equivalence after inverting \(h\). Compatibility between odd and even theories modulo \(2\) supplies the required commutative square.

Thus a concordance produces local maps in both directions:
\[
LEO(K_0)\longrightarrow LEO(K_1),
\qquad
LEO(K_1)\longrightarrow LEO(K_0).
\]
The assignment
\[
\Phi_{\mathrm{LEO}}:\mathcal C\longrightarrow C_{\mathrm{LEO}},
\qquad
[K]\longmapsto[LEO(K)]
\]
is a group homomorphism. If \(K\) is slice, then
\[
[LEO(K)]=[LEO(U)].
\]

The construction retains the odd Khovanov complex, the Bar–Natan-deformed even complex, the mod-\(2\) odd/even identification, the \(X\)-module structures, and the localized Bar–Natan complex. Its local-equivalence class determines Rasmussen’s \(s\)-invariant but also retains integral torsion, Bockstein operations, odd/even compatibility, and the behavior of classes under localization.

The Bar–Natan-only construction is Lewark’s group \(G(\mathbb Z[h])\). A knot-like complex \(D\) over a graded \(\mathbb Z[h]\)-algebra satisfies
\[
h^{-1}D\simeq h^{-1}\mathbb Z[h],
\]
and a chain map is local if it becomes a chain homotopy equivalence after inverting \(h\). Local-equivalence classes form an abelian group under tensor product, with dualization as inverse. The reduced Bar–Natan complex defines
\[
S:\mathcal C\longrightarrow G(\mathbb Z[h]).
\]

The universal Khovanov construction gives a related but distinct target. For
\[
R=\mathbb Z[G],\qquad q\deg G=-2,
\]
the reduced universal Khovanov complex \(C(D)\) is a finite chain complex of graded-free \(R\)-modules. It is obtained from Bar–Natan’s cobordism category or from
\[
A=\mathbb Z[G,X]/(X^2+GX)
\]
reduced at \(X=0\). Specializing \(G=1\) gives homology \(\mathbb Z\) in degree \(0\) and zero elsewhere. Connected sum corresponds to tensor product, mirroring to dualization, and genus-\(g\) cobordisms induce maps of quantum degree \(-2g\).

Two admissible complexes are \(\mathbb Z\)-equivalent if there are degree-zero maps in both directions inducing isomorphisms on
\[
H_0(-\otimes_{\mathbb Z[G]}\mathbb Z).
\]
The resulting knot-like classes form an abelian group \(\mathcal G\), and
\[
S(K)=[C(D)]
\]
defines a concordance homomorphism
\[
S:\mathcal C\longrightarrow\mathcal G.
\]
Its group operation is induced by tensor product, and
\[
S(-K)=S(K)^*=S(K)^{-1}.
\]
A slice knot maps to the identity.

## 4. Refined \(s\)-invariants and triviality detection

Let \(\mathbb F\) be a field. For a LEO triple, localized homology has two free summands together with \(h\)-torsion:
\[
H(D\otimes\mathbb F)
\cong
\mathbb F[h]\oplus\mathbb F[h]\oplus T.
\]
The field-valued invariants \(s^+_{\mathbb F}\) and \(s^-_{\mathbb F}\) measure the highest quantum grading in which localized classes are respectively nonzero or surjective. For reduced triples, the invariant is
\[
s_{\mathbb F}(C,D,f)
=
\max\left\{
q\ \middle|\
H^{0,q}(D;\mathbb F)\to H^{0,q}(h^{-1}D;\mathbb F)
\text{ is nonzero}
\right\}.
\]
For knots,
\[
s^+_{\mathbb F}(LEO(K))
=
s^-_{\mathbb F}(LEO(K))
=
s_{\mathbb F}(LEO(K)),
\]
and this equals Rasmussen’s invariant over \(\mathbb F\). The reduced invariant is additive under tensor product and therefore defines a homomorphism on the appropriate local-equivalence group.

Integral refinements use the filtration on localized homology. For a reduced triple,
\[
s_{\mathbb Z}(C,D,f)
=
\max\left\{
q\ \middle|\
H^{0,q}(D;\mathbb Z)\to H^{0,q}(h^{-1}D;\mathbb Z)
\text{ is surjective}
\right\}.
\]
The integral upper invariant \(s^+_{\mathbb Z}\) records the highest grading containing a class whose localized image is primitive. The graded length is
\[
\operatorname{gl}(C,D,f)
=
\frac{s^+_{\mathbb Z}(C,D,f)-s_{\mathbb Z}(C,D,f)}{2}.
\]
The cyclic groups
\[
H^0(h^{-1}D)^{(q)}
\]
are local-equivalence invariants, and their orders define the integral refined invariant
\[
s^{\mathbb Z}(C,D,f)
=
\left(
s^+_{\mathbb Z},
c(s^+_{\mathbb Z}-2),
\dots,
c(s^+_{\mathbb Z}-2\operatorname{gl})
\right).
\]
For knots, these recover the information in Schütz’s integral \(s\)-invariant.

Bockstein refinements arise from
\[
0\longrightarrow \mathbb Z/2
\longrightarrow \mathbb Z/2^{n+1}
\longrightarrow \mathbb Z/2^n
\longrightarrow0.
\]
The resulting maps
\[
\beta_n:H^k(C;\mathbb Z/2^n)\longrightarrow H^{k+1}(C;\mathbb F_2)
\]
define the invariants \(r^{\beta_n}\) and \(s^{\beta_n}\). For \(n=1\),
\[
\beta=\beta_1+f^{-1}\circ\beta_1\circ f,
\]
which, for knots, is the sum of the even and odd first Steenrod squares:
\[
Sq^1_e+Sq^1_o.
\]
The invariants descend under local equivalence, and for knots the successive refinements satisfy
\[
s^{\beta_1}(K)\leq s^{\beta_2}(K)\leq\cdots\leq s^{\beta_\infty}(K).
\]

The comprehensive invariants \(r_o,s_o\) and \(r_c,s_c\) incorporate, respectively, mod-\(2\) odd compatibility and integral primitive-lifting conditions. For knots,
\[
s_{\mathbb Z}(K)-2
\leq
r_c(K)=s_c(K)
\leq
s_{\mathbb Z}(K),
\]
and
\[
s_{\mathbb F_2}(K)-2
\leq
r_o(K)=s_o(K)
\leq
s_{\mathbb F_2}(K),
\]
with
\[
s_c(K)\leq s_o(K).
\]
These comprehensive invariants are stronger sliceness obstructions than the individual Bockstein refinements.

For reduced LEO triples, triviality is detected by
\[
(C,D,f)=0
\quad\Longleftrightarrow\quad
s_c(C,D,f)=0
\quad\text{and}\quad
s_c(C^*,D^*,f^*)=0.
\]
In the two-reduced theory, the analogous criterion uses \(s_o\). For knots,
\[
r_c(LEO(K))=r_c(LEO(\overline K))=0
\]
implies
\[
LEO(K)\sim LEO(U).
\]
Thus the pair of comprehensive invariants evaluated on a knot and its mirror detects whether its LEO class is trivial.

## 5. Computations and sliceness obstructions

Computations implemented in KnotJob compare \(s^{\beta_{15}}\), \(s^\beta\), and
\[
\bigl(s_c(LEO(K)),-s_c(LEO(\overline K))\bigr)
\]
with the earlier invariants
\[
s^{Sq^1_e},\qquad
s^{Sq^1_o},\qquad
s^\beta,\qquad
s^{\beta_{15}}.
\]
For prime knots through \(16\) crossings, the refined invariants are frequently nonconstant among knots sharing the same ordinary \(s_{\mathbb F_2}\)-value. The number of such knots grows substantially with crossing number, and the comprehensive \(s_c\)-data provide the largest counts in the reported comparison.

Among \(17{,}991\) prime knots with at most \(19\) crossings whose sliceness remained unresolved in the cited Dunfield–Gong data, \(826\) were obstructed by \(s^{\beta_{15}}\), \(64\) by \(s^\beta\), and \(890\) had nonzero \(s_c\)-data. The \(s^{\beta_{15}}\)- and \(s^\beta\)-detected sets were disjoint, and \(s_c\) was nonzero exactly when one of those two invariants was nonzero. The refinements therefore obstructed sliceness for \(890\) knots, reducing the mystery-knot list by about \(5\%\). Many of these knots have Alexander polynomial \(1\), so they are topologically slice; the resulting obstructions concern smooth sliceness.

Manolescu and Piccirillo exhibited five topologically slice knots whose smooth sliceness would imply the existence of an exotic \(4\)-sphere. Nakamura proved that these knots are not smoothly slice using \(0\)-surgery homeomorphisms and stable concordance arguments. The Khovanov local-equivalence analysis gives an independent obstruction: the ordinary \(s\)-invariant, the even and odd \(Sq^1\)-refinements, and \(s^{\beta_{15}}\) vanish, whereas the sum refinement \(s^\beta\) is nonzero for all five knots. This demonstrates that the simultaneous interaction of even and odd Khovanov Bocksteins can contain information absent from either theory separately [2312.09114].

The universal integral Khovanov homomorphism has a different computational strength. After extension to a field \(\mathbb F\), Smith normal form decomposes the universal complex into contractible pieces, a rank-one free summand
\[
t^0q^{s_c(K)}\mathbb F[G],
\]
and torsion-type summands. The distinguished grading \(s_c(K)\) is the Rasmussen invariant in the relevant characteristic. The universal class determines every \(s_c\) and Schütz’s \(s^{\mathbb Z}\), but it also retains integral divisibility and extension data discarded by individual field reductions.

Staircase complexes
\[
{}_{(a_1,\ldots,a_n)}
\]
encode divisibility sequences and tensor multiplicative relations. The subgroup generated by staircase classes is free abelian of infinite rank:
\[
S_{\mathrm{staircase}}\cong\mathbb Z^{\oplus\infty}.
\]
The family
\[
C^k={}_{(2^k)}\otimes{}_{(2^{k+1})}^*
\]
has vanishing Rasmussen invariants in every characteristic and identical Schütz invariant
\[
s^{\mathbb Z}(C^k)=(0,2),
\]
yet its members are pairwise non-equivalent. Hence the universal integral target is strictly stronger than the combined collection of field-valued Rasmussen invariants [2401.08480].

## 6. Structure beyond Rasmussen invariants

The later structure theory separates the Bar–Natan group from the additional odd/even information. For a graded \(\mathbb Z[h]\)-algebra \(R\), Lewark’s group \(G(R)\) consists of local-equivalence classes of knot-like complexes. The even–odd group \(\mathcal{LEO}(Q,R)\) consists of triples \((C,D,f)\), with \(D\) knot-like and \(f\) identifying the reductions of \(C\) and \(D\).

Projection to the Bar–Natan component is
\[
\mathcal{LEO}(Q,R)\longrightarrow G(R),
\qquad
(C,D,f)\longmapsto D.
\]
The kernel measures information invisible on the Bar–Natan side. It is already large: the kernel of
\[
\mathcal{LEO}\longrightarrow G(\mathbb Z[h])
\]
contains a subgroup isomorphic to
\[
(\mathbb Z/2\mathbb Z)^\infty
\]
and a \(\mathbb Z^\infty\) direct summand [2610.37595].

For a prime \(p\), explicit knot-like complexes \(D_{p,m,n}\) over \(\mathbb Z[h]\) generate independent \(2\)-torsion classes. Their distinction is detected by \(p\)-adic local-equivalence invariants. Over \(\mathbb Z_p[h]/(ph)\), knot-like complexes with \(s_{\mathbb Z_p}=0\) are classified by standard complexes
\[
C(a_1,\ldots,a_{2n}),
\]
whose successive differentials alternate between powers of \(p\) and powers of \(h\). The local order is lexicographic in the reciprocal sequences
\[
(1/a_1,\ldots,1/a_{2n}).
\]

The integer-valued coordinates
\[
\varphi_j(C(a_1,\ldots,a_{2n}))
=
\#\{i\text{ odd}:a_i=j\}
-
\#\{i\text{ odd}:a_i=-j\},
\]
and
\[
\psi_j(C(a_1,\ldots,a_{2n}))
=
\#\{i\text{ even}:a_i=j\}
-
\#\{i\text{ even}:a_i=-j\}
\]
are homomorphisms. They detect a \(\mathbb Z^\infty\) direct summand in the kernel of the Bar–Natan projection, and they are invisible to all field-valued Rasmussen invariants.

At \(p=2\), an identification
\[
\mathcal{LEO}(\mathbb F_2,\mathbb Z_2[h])
\cong
G(\mathbb Z_2[h]/(2h))
\]
transfers this structure to the odd/even local-equivalence group. Composing with the concordance map yields integer-valued concordance homomorphisms
\[
\varphi_j^o=\varphi_j\circ LEO,
\qquad
\psi_j^o=\psi_j\circ LEO.
\]
The six homomorphisms
\[
\varphi_1^o,\ \varphi_2^o,\ \varphi_3^o,\ \varphi_4^o,\ \varphi_5^o,\ \psi_1^o
\]
are linearly independent from one another, from all Rasmussen invariants in all characteristics, from Ozsváth–Szabó’s \(\tau\), and from the Dai–Hom–Stoffregen–Truong concordance homomorphisms. Their independence was established by evaluating them on knots through \(16\) crossings and row-reducing the resulting matrix [2610.37595].

The resulting hierarchy is therefore broader than a collection of numerical \(s\)-invariants. The Bar–Natan group retains \(p\)-adic and \(h\)-adic structure, while the LEO group additionally records the gluing of odd Khovanov data to the even Bar–Natan reduction. Its kernel contains both torsion and free-rank information invisible to the Bar–Natan component.

## 7. Scope and related local constructions

Khovanov local equivalence should be distinguished from local reductions in Khovanov–Rozansky operator formalisms. In the latter setting, local operators are assigned to MOY resolutions, and local conjugations, quasi-isomorphisms, Reidemeister equivalences, and elimination of contractible summands relate complexes in a resolution hypercube. These operations produce equivalences of vertical or horizontal complexes under specified hypotheses, but they do not define a group of local transformations with a presentation, composition law, and inverse for every reduction [2605.01584].

The Khovanov–Rozansky construction uses a bicomplex. Vertical cohomology is computed from local differential operators attached to individual MOY resolutions; horizontal cohomology is then computed from maps between those vertical cohomologies. For closed diagrams, the relevant operator is nilpotent because incoming and outgoing boundary contributions to the superpotential cancel. Local reductions preserve the global invariant when they are chain isomorphisms, quasi-isomorphisms, homotopy equivalences, or eliminations of contractible summands compatible with the horizontal complex.

This operator-theoretic locality differs from the algebraic local equivalence defining \(C_{\mathrm{LEO}}\), \(G(\mathbb Z[h])\), and \(\mathcal G\). In Khovanov local-equivalence groups, the central operation is passage to equivalence classes under maps that are invertible only on a selected localized summand. In the Khovanov–Rozansky setting, locality describes the support of operators and reductions within the resolution hypercube, while the resulting cohomology remains a global invariant of the full diagram.

The combined picture is:
\[
\text{odd Khovanov}
+
\text{Bar--Natan deformation}
+
\text{mod-2 identification}
\Longrightarrow
\text{LEO local-equivalence group}.
\]
Connected sum induces tensor product, mirrors induce dualization, the unknot represents the identity, and concordances induce local maps. Rasmussen’s invariant records the extremal grading of localized generators; refined local-equivalence invariants additionally record integral lifting, torsion, Bockstein operations, even/odd interaction, graded length, and dual behavior. The universal integral construction and the even–odd LEO construction consequently exhibit Khovanov concordance information substantially richer than the field-valued \(s\)-invariants alone.

Source: https://www.emergentmind.com/topics/khovanov-local-equivalence-groups