---
title: Persistent Khovanov Homology
url: https://www.emergentmind.com/topics/persistent-khovanov-homology
type: topic
---

# Persistent Khovanov Homology

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Persistent Khovanov homology of tangles is a framework in knot data analysis (KDA) for characterizing local topological features in curve-type data such as knots, links, and tangles. It is introduced to address a limitation of evolutionary Khovanov homology, which is fundamentally a global invariant and hence insufficient to resolve local, spatially localized topological features within a curve. The construction starts from Bar-Natan’s bracket complex, introduces a concrete functor from the category of tangles to modules so that the Khovanov complex can be computed in the abelian category of modules, and uses planar algebra to build a category of tangles without fixed boundaries, thereby enabling persistence for locally evolving tangles [2409.18312].

## 1. Motivation and scope

Knot Data Analysis studies curve-type data arising in applications such as river paths, vascular networks, DNA/RNA, and materials [2409.18312]. In this setting, standard persistent homology on simplicial or cubical complexes reveals multiscale Betti numbers but does not naturally encode the homotopic, link-theoretic, and tangle-theoretic features of curve-type data. The multiscale Gauss link integral captures multiscale information but does not preserve topological invariants at small scales. Evolutionary Khovanov homology extends Khovanov homology to multiscale analysis of links, but it is fundamentally a global invariant and hence insufficient to resolve local features such as localized tangle patterns [2409.18312].

A further obstacle is categorical. Existing categories of tangles typically require fixed boundaries throughout the filtration, which is unrealistic in many data-driven scenarios where the window or region grows and endpoints move. Persistent Khovanov homology of tangles is designed to address this issue by introducing persistence both with fixed boundary and without fixed boundary, the latter via planar algebra morphisms that model local growth of a tangle region [2409.18312].

The resulting framework has two stated contributions. First, a concrete functor from the category of tangles to modules is constructed so that the Khovanov complex can be computed in the abelian category of modules. Second, a new category of tangles without fixed boundaries is built using planar algebra, with morphisms given by inclusions via 1-input planar tangles, enabling persistent Khovanov homology for locally evolving tangles [2409.18312].

## 2. Categorical and algebraic construction

Let $B$ be a finite set of points on a circle. The Bar-Natan framework uses cobordism categories in which the objects are tangles in a disk $D$ with boundary $B$, while morphisms are $2$D cobordisms between tangles inside $D \times [-\epsilon,\epsilon] \times [0,1]$, fixed on $B \times [-\epsilon,\epsilon] \times [0,1]$. This yields the $4$D cobordism category $\mathrm{Cob}^4(B)$. For smoothing data, the construction uses $\mathrm{Cob}^3(B)$, its pre-additive version $k\mathrm{Cob}^3(B)$, the localization $k\mathrm{Cob}^3_{/l}(B)$ by the relations $(S)$, $(T)$, and $(4Tu)$, and finally the additivization $\mathrm{Mat}(k\mathrm{Cob}^3_{/l}(B))$ [2409.18312].

To remove the fixed-boundary requirement, the paper introduces the category $\mathrm{Pla}$. Its objects are tangles with no fixed boundary set $B$ enforced. Its morphisms $T \to T'$ are induced by a $1$-input planar tangle $D$, with $T' = D(T)$. Concretely, a morphism is an inclusion of $1$-manifolds: arcs map to arcs or circles, and circles map to circles. This is the categorical mechanism that models local growth of a tangle region without fixing the endpoints on a common boundary throughout the filtration [2409.18312].

Two module-valued functors are central.

For links, the standard TQFT functor is
$$
F:\mathrm{Cob}^3(\varnothing)\to \mathrm{Mod}_k,
$$
where $k$ is a commutative ring with unit and $V = k\{v_+,v_-\}$. For a link $L$ with $r(L)$ circles,
$$
F(L)=V^{\otimes r(L)}.
$$
On generating cobordisms,
$$
F(\cap):k\to V,\quad 1\mapsto v_+,
$$
$$
F(\cup):V\to k,\quad v_+\mapsto 0,\; v_-\mapsto 1,
$$
and for saddle cobordisms,
$$
F(\wedge)=\Delta:V\to V\otimes_k V,
$$
with
$$
\Delta(v_+) = v_+ \otimes v_- + v_- \otimes v_+,\qquad \Delta(v_-) = v_- \otimes v_-,
$$
while
$$
F(\vee)=m:V\otimes_k V\to V,
$$
with
$$
m(v_+\otimes v_+) = v_+,\quad m(v_-\otimes v_+) = v_-,\quad m(v_+\otimes v_-) = v_-,\quad m(v_-\otimes v_-) = 0.
$$
This extends to a functor $F:k\mathrm{Cob}^3_{/l}(\varnothing)\to \mathrm{Mod}_k$ and then to chain complexes of modules via composition with $Kh_\varnothing$ [2409.18312].

For tangles with arcs, the paper introduces
$$
G:\mathrm{Cob}^3(B)\to \mathrm{Mod}_k.
$$
Let $V = k\{v_+,v_-\}$ and $W = k\{w\}$. For a tangle $T$ with $r(T)$ circles and $t(T)$ arcs,
$$
G(T)=W^{\otimes t(T)}\otimes_k V^{\otimes r(T)}.
$$
On cobordisms involving arcs,
- the saddle of two independent arcs is zero on $W\otimes W$;
- the local saddle that creates a circle from an arc is
  $$
  W\to W\otimes V,\qquad w\mapsto w\otimes v_-;
  $$
- the local saddle that merges a circle into an arc is
  $$
  W\otimes V\to W,\qquad w\otimes v_+\mapsto w,\quad w\otimes v_-\mapsto 0.
  $$
On closed-circle operations, $G$ coincides with $F$, hence uses the same $\Delta$ and $m$. The grading conventions are
$$
\deg(v_+) = 1,\qquad \deg(v_-) = -1,\qquad \deg(w) = -1.
$$
Because the relations $(S)$, $(T)$, and $(4Tu)$ occur on closed components, $G$ descends to
$$
G:k\mathrm{Cob}^3_{/l}(B)\to \mathrm{Mod}_k
$$
and extends to complexes via $G^\bullet$ [2409.18312].

## 3. Bracket complex, Khovanov complex, and gradings

For a tangle diagram $T$ with $n$ crossings and $n_-$ left-handed crossings, each state $s\in\{0,1\}^n$ determines a smoothing $T_s$ with
$$
h(s)=\ell(s)-n_-,\qquad \ell(s)=\sum s_i.
$$
The bracket complex is
$$
[[T]]^k=\bigoplus_{h(s)=k} T_s,
$$
with differential
$$
d^k=\sum_\xi (-1)^{\mathrm{sgn}(\xi)} d_\xi:[[T]]^k\to [[T]]^{k+1},
$$
where $\xi$ runs over cube edges changing a single smoothing $0\to 1$ and $d_\xi$ is the saddle cobordism between $T_s$ and $T_{s'}$. The cube is anti-commutative: on each square face,
$$
d_{\tilde\xi}\circ d_\eta = - d_{\eta'}\circ d_\xi.
$$
Accordingly, $([[T]]^\ast,d^\ast)$ is a cochain complex in $\mathrm{Mat}(k\mathrm{Cob}^3(\partial T))$, and after passing to $k\mathrm{Cob}^3_{/l}(\partial T)$ it becomes a chain-homotopy invariant [2409.18312].

The Khovanov complex is obtained by the writhe shift
$$
Kh^p(T) = [[T]]^{p+n_+-n_-},\qquad d_T^p = d^{p+n_+-n_-}.
$$
The quantum grading on an element $x$ in a cochain group is
$$
\Phi(x)=p+n_+-n_-+\theta(x),
$$
where
$$
\theta(v_+)=1,\qquad \theta(v_-)=-1,
$$
and for the functor $G$ also
$$
\theta(w)=-1.
$$
After applying a module-valued functor $\mathfrak{F}\in\{F,G\}$, the differential becomes
$$
d^k=\sum_\xi (-1)^{\mathrm{sgn}(\xi)}\mathfrak{F}(d_\xi),
$$
with identity on components unaffected by the saddle. On closed components the algebraic data are the Frobenius algebra operations $(A,\Delta,m,\epsilon,\eta)$ with $A=V$; for tangles with arcs, the additional arc operations are exactly those encoded by $G$ [2409.18312].

If $\partial T=\varnothing$, then $H^p(T;\mathcal{G})=H^p(T;\mathcal{F})$, so the construction recovers classical Khovanov homology. The Euler characteristic recovers the unnormalized Jones polynomial in the link case [2409.18312].

## 4. Persistent structure and persistence modules

With fixed boundary, a persistence tangle with boundary $B$ is a functor
$$
\mathcal{P}:(X,\le)\to \mathrm{Cob}^4(B).
$$
Its persistent Khovanov homology, with respect to a functor $\mathfrak{F}$ to modules and homology $H$, is the composition
$$
(X,\le)\xrightarrow{\mathcal{P}} \mathrm{Cob}^4(B)\xrightarrow{H(-;\mathfrak{F})}\mathrm{Mod}_k.
$$
For any $a\le b\in X$ and any $p\in\mathbb{Z}$, the $(a,b)$-persistent Khovanov homology is
$$
H^p_{a,b}(\mathcal{P},B)=\big(H^p(\mathcal{P}(a),B)\to H^p(\mathcal{P}(b),B)\big),
$$
with Betti polynomial
$$
\beta^p_{a,b}(q)=\sum_{\omega\in H^p_{a,b}} q^{\Phi(\omega)}.
$$
This is the fixed-boundary persistence theory described in the paper [2409.18312].

Without fixed boundary, a persistence tangle in $\mathrm{Pla}$ is a functor
$$
\mathcal{P}:(X,\le)\to \mathrm{Pla},
$$
and its persistent Khovanov homology is
$$
(X,\le)\xrightarrow{\mathcal{P}} \mathrm{Pla}\xrightarrow{H(-;\mathcal{G})}\mathrm{Mod}_k.
$$
For $a\le b$,
$$
H^p_{a,b}(\mathcal{P};\mathcal{G})=\big(H^p(\mathcal{P}(a);\mathcal{G})\to H^p(\mathcal{P}(b);\mathcal{G})\big).
$$
The essential point is that the chain-level construction per object remains the standard Bar-Natan bracket/Khovanov complex; persistence is implemented by functorial maps between the chain complexes induced by planar-algebra inclusions [2409.18312].

When $(X,\le)=(\mathbb{Z},\le)$, the direct sum
$$
\mathbf{H}=\bigoplus_{a\in\mathbb{Z}} H^\ast(\mathcal{P}(a),B)
$$
has a natural $k[z]$-module structure via the shift map
$$
z:H^\ast(\mathcal{P}(a),B)\to H^\ast(\mathcal{P}(a+1),B).
$$
The same applies in the planar setting with $H^\ast(-;\mathcal{G})$. The paper remarks that under standard assumptions, the structure theorem and stability results for persistence modules apply, though details are not elaborated [2409.18312].

## 5. Functoriality, invariance, and induced maps

Bar-Natan invariance gives the starting point: the bracket complex is invariant up to chain homotopy in $k\mathrm{Cob}^3_{/l}(\partial T)$, and hence $Kh^\ast(T)$ is a tangle invariant up to chain homotopy. The functor
$$
Kh_B:\mathrm{Cob}^4(B)\to Ch^\bullet(\mathrm{Mat}(k\mathrm{Cob}^3_{/l}(B)))
$$
maps isotopy classes of tangles to chain-homotopy classes [2409.18312].

For any additive module-valued functor
$$
\mathfrak{F}:k\mathrm{Cob}^3_{/l}(B)\to Ab,
$$
one obtains
$$
\mathfrak{F}^\bullet\circ Kh_B:\mathrm{Cob}^4(B)\to Ch^\bullet(Ab),
$$
and the resulting homology is an isotopy invariant. In particular, Theorem 3.2 states that
$$
G^\bullet Kh_B:\mathrm{Cob}^4(B)\to Ch^\bullet(\mathrm{Mod}_k)
$$
maps isotopy classes of tangles to homotopy classes of cochain complexes [2409.18312].

The paper also gives explicit induced maps for generators of morphisms in $\mathrm{Cob}^4(\varnothing)$. For the cap cobordism
$$
\cap:T\to T\sqcup \bigcirc,
$$
the induced map is
$$
K_\varnothing(\cap):K_\varnothing(T)\to K_\varnothing(T)\otimes V,\qquad x\mapsto x\otimes v_+,
$$
hence
$$
\mathrm{im}\,H^\ast(\cap;\mathfrak{F}) = H^\ast(T;\mathfrak{F})\otimes v_+.
$$
For the cup cobordism
$$
\cup:T\sqcup \bigcirc\to T,
$$
one has
$$
K_\varnothing(\cup)(x\otimes v_+)=0,\qquad K_\varnothing(\cup)(x\otimes v_-)=x,
$$
hence
$$
H^\ast(\cup;\mathfrak{F}) = H^\ast(T;\mathfrak{F}).
$$
For a local saddle $T\to T'$, the map is described using the mapping-cone description via the crossing-change tangle $\widetilde{T}$ and the cochain map
$$
K_\varnothing(\mathrm{saddle}) = p_1\circ \tilde d,
$$
which induces
$$
(p_1\tilde d)^\ast:H^\ast(T;\mathfrak{F})\to H^\ast(T';\mathfrak{F}).
$$
These formulas enable step-by-step computation of persistent Khovanov homology along a filtration [2409.18312].

In the planar-algebra category $\mathrm{Pla}$, the paper constructs a cochain map
$$
\Psi:GKh^\ast(T)\to GKh^\ast(T'),
$$
for morphisms $T\to T'=D(T)$ induced by a $1$-input planar tangle $D$. The map acts on independent components as identity on arc $\to$ arc and circle $\to$ circle, while on arc $\to$ circle it is
$$
\Psi:G(\mathrm{arc})\to G(\mathrm{circle}),\qquad w\mapsto v_-.
$$
The chain-map property is verified by commuting diagrams with $m$ and $\Delta$, and the construction
$$
G^\bullet Kh:\mathrm{Pla}\to Ch^\bullet(\mathrm{Mod}_k)
$$
is functorial. However, there is no notion of isotopy between tangles with different boundaries, so no isotopy invariance statement is claimed in $\mathrm{Pla}$ [2409.18312].

## 6. Computation, examples, applications, and limitations

The computational framework is the usual $2^n$-state construction. Given a tangle diagram $T$ with $n$ crossings, one builds the cube of states $\{0,1\}^n$, forms each smoothing $T_s$, assembles the bracket complex
$$
[[T]]^k=\bigoplus_{h(s)=k}T_s,
$$
shifts to
$$
Kh^p(T)=[[T]]^{p+n_+-n_-},
$$
applies $G$ or $F$, and computes homology with quantum grading
$$
\Phi(x)=p+n_+-n_-+\theta(x).
$$
For $G$, each smoothing with $t$ arcs and $r$ circles contributes
$$
G(T_s)=W^{\otimes t}\otimes V^{\otimes r},
$$
and each differential is a sum of local saddle maps with identity elsewhere [2409.18312].

The paper presents three small tangle examples. If $T$ is a single left-handed crossing on an arc, the cochain complex collapses to
$$
0\to Kh^{-1}(T)=\mathrm{arc}\xrightarrow{\mathrm{saddle}} Kh^0(T)=\mathrm{arc}\sqcup \mathrm{circle}\to 0.
$$
Applying $G$ gives
$$
0\to W\xrightarrow{d} W\otimes V\to 0,\qquad d(w)=w\otimes v_-.
$$
Hence
$$
H^0(T;\mathcal{G})\cong k\{w\otimes v_+\},\qquad H^p=0\ \text{for}\ p\neq 0,
$$
and
$$
\Phi(w\otimes v_+)=-1.
$$
If $T'$ is a single right-handed crossing on an arc, then
$$
0\to Kh^0(T')=\mathrm{arc}\sqcup \mathrm{circle}\xrightarrow{\mathrm{saddle}} Kh^1(T')=\mathrm{arc}\to 0,
$$
and after applying $G$,
$$
0\to W\otimes V\xrightarrow{d} W\to 0,\qquad d(w\otimes v_+)=w,\quad d(w\otimes v_-)=0.
$$
Hence
$$
H^0(T';\mathcal{G})\cong k\{w\otimes v_-\},\qquad H^p=0\ \text{for}\ p\neq 0,
$$
with
$$
\Phi(w\otimes v_-)=-1.
$$
If $T''$ is a single arc with no crossings, then
$$
H^0(T'';\mathcal{G})\cong k\{w\},\qquad H^p=0\ \text{for}\ p\neq 0,
$$
and
$$
\Phi(w)=-1.
$$
These three tangles are equivalent up to Reidemeister moves, and their Khovanov homology groups and gradings of generators agree [2409.18312].

The applications emphasized in the paper are in KDA. For a planar tangle $T$ and a center $P$, one may define a single-center radial filtration by
$$
T_\epsilon = T\cap D_\epsilon,
$$
where $D_\epsilon$ is the disk of radius $\epsilon$ centered at $P$. Then
$$
\mathcal{P}:(\mathbb{R},\le)\to \mathrm{Pla},\qquad \mathcal{P}(\epsilon)=T_\epsilon,
$$
is a persistence tangle, and one computes
$$
H^p_{a,b}(\mathcal{P};\mathcal{G})
$$
to capture birth and death of local features around $P$ as the window grows. For a finite collection of curves $C\subset \mathbb{R}^3$ and a generic projection $q:C\to\mathbb{R}^2$ with only double crossings, the same construction using
$$
T_\epsilon=q(C)\cap D_\epsilon
$$
again defines a persistence tangle in $\mathrm{Pla}$ and yields persistent Khovanov homology of tangles as multiscale local descriptors [2409.18312].

Several limitations are explicitly identified. The approach inherits the $2^n$ scaling of state-cube constructions in Khovanov theory, and explicit complexity bounds and specific data structures are not developed. In the varying-boundary category $\mathrm{Pla}$, the construction is functorial but not an isotopy invariant, because there is no notion of isotopy between tangles with different boundaries. The paper notes that under certain conditions the standard structure theorem and stability for persistence modules carry over, but it does not develop proofs, bounds, or metrics such as interleaving distances tailored to persistent Khovanov homology of tangles. It also does not analyze robust preprocessing for noisy, discretized curves, nor the choice of projections and centers in the planar filtration examples [2409.18312].

Persistent Khovanov homology of tangles therefore occupies a distinct position relative to both standard persistent homology and evolutionary Khovanov homology. Its base invariant is Khovanov homology of tangles, a categorification of the Jones polynomial, and its stated advantages are retention of link- and tangle-theoretic information, local analysis via inclusions induced by planar tangles, and explicit algebraic formulas for maps associated to $\cap$, $\cup$, saddle, and $\Psi$. Its stated tradeoffs are more complex chain complexes, lack of an isotopy-invariance statement in $\mathrm{Pla}$, and growth of module sizes with the numbers of arcs and circles [2409.18312].

Source: https://www.emergentmind.com/topics/persistent-khovanov-homology