Persistent Khovanov Homology

• Persistent Khovanov homology is a framework that generalizes standard Khovanov homology to resolve local topological features in tangles and links.
• It employs Bar-Natan’s bracket complex and a functor from the tangle category to modules, using planar algebra to relax fixed-boundary constraints.
• The method advances knot data analysis by providing explicit algebraic maps for local changes and capturing multiscale persistence in curve-type data.

Searching arXiv for the cited paper and closely related work to ground the article. 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 (Liu et al., 2024).

1. Motivation and scope

Knot Data Analysis studies curve-type data arising in applications such as river paths, vascular networks, DNA/RNA, and materials (Liu et al., 2024). 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 (Liu et al., 2024).

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 (Liu et al., 2024).

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 (Liu et al., 2024).

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 $D$0 by the relations $D$1, $D$2, and $D$3, and finally the additivization $D$4 (Liu et al., 2024).

To remove the fixed-boundary requirement, the paper introduces the category $D$5. Its objects are tangles with no fixed boundary set $D$6 enforced. Its morphisms $D$7 are induced by a $D$8-input planar tangle $D$9, with $B$0. Concretely, a morphism is an inclusion of $B$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 (Liu et al., 2024).

Two module-valued functors are central.

For links, the standard TQFT functor is

$B$2

where $B$3 is a commutative ring with unit and $B$4. For a link $B$5 with $B$6 circles,

$B$7

On generating cobordisms,

$B$8

$B$9

and for saddle cobordisms,

$2$0

with

$2$1

while

$2$2

with

$2$3

This extends to a functor $2$4 and then to chain complexes of modules via composition with $2$5 (Liu et al., 2024).

For tangles with arcs, the paper introduces

$2$6

Let $2$7 and $2$8. For a tangle $2$9 with $D \times [-\epsilon,\epsilon] \times [0,1]$0 circles and $D \times [-\epsilon,\epsilon] \times [0,1]$1 arcs,

$D \times [-\epsilon,\epsilon] \times [0,1]$2

On cobordisms involving arcs,

• the saddle of two independent arcs is zero on $D \times [-\epsilon,\epsilon] \times [0,1]$3;
• the local saddle that creates a circle from an arc is

$D \times [-\epsilon,\epsilon] \times [0,1]$4

• the local saddle that merges a circle into an arc is

$D \times [-\epsilon,\epsilon] \times [0,1]$5

On closed-circle operations, $D \times [-\epsilon,\epsilon] \times [0,1]$6 coincides with $D \times [-\epsilon,\epsilon] \times [0,1]$7, hence uses the same $D \times [-\epsilon,\epsilon] \times [0,1]$8 and $D \times [-\epsilon,\epsilon] \times [0,1]$9. The grading conventions are

$B \times [-\epsilon,\epsilon] \times [0,1]$0

Because the relations $B \times [-\epsilon,\epsilon] \times [0,1]$1, $B \times [-\epsilon,\epsilon] \times [0,1]$2, and $B \times [-\epsilon,\epsilon] \times [0,1]$3 occur on closed components, $B \times [-\epsilon,\epsilon] \times [0,1]$4 descends to

$B \times [-\epsilon,\epsilon] \times [0,1]$5

and extends to complexes via $B \times [-\epsilon,\epsilon] \times [0,1]$6 (Liu et al., 2024).

3. Bracket complex, Khovanov complex, and gradings

For a tangle diagram $B \times [-\epsilon,\epsilon] \times [0,1]$7 with $B \times [-\epsilon,\epsilon] \times [0,1]$8 crossings and $B \times [-\epsilon,\epsilon] \times [0,1]$9 left-handed crossings, each state $4$0 determines a smoothing $4$1 with

$4$2

The bracket complex is

$4$3

with differential

$4$4

where $4$5 runs over cube edges changing a single smoothing $4$6 and $4$7 is the saddle cobordism between $4$8 and $4$9. The cube is anti-commutative: on each square face,

$\mathrm{Cob}^4(B)$0

Accordingly, $\mathrm{Cob}^4(B)$1 is a cochain complex in $\mathrm{Cob}^4(B)$2, and after passing to $\mathrm{Cob}^4(B)$3 it becomes a chain-homotopy invariant (Liu et al., 2024).

The Khovanov complex is obtained by the writhe shift

$\mathrm{Cob}^4(B)$4

The quantum grading on an element $\mathrm{Cob}^4(B)$5 in a cochain group is

$\mathrm{Cob}^4(B)$6

where

$\mathrm{Cob}^4(B)$7

and for the functor $\mathrm{Cob}^4(B)$8 also

$\mathrm{Cob}^4(B)$9

After applying a module-valued functor $\mathrm{Cob}^3(B)$0, the differential becomes

$\mathrm{Cob}^3(B)$1

with identity on components unaffected by the saddle. On closed components the algebraic data are the Frobenius algebra operations $\mathrm{Cob}^3(B)$2 with $\mathrm{Cob}^3(B)$3; for tangles with arcs, the additional arc operations are exactly those encoded by $\mathrm{Cob}^3(B)$4 (Liu et al., 2024).

If $\mathrm{Cob}^3(B)$5, then $\mathrm{Cob}^3(B)$6, so the construction recovers classical Khovanov homology. The Euler characteristic recovers the unnormalized Jones polynomial in the link case (Liu et al., 2024).

4. Persistent structure and persistence modules

With fixed boundary, a persistence tangle with boundary $\mathrm{Cob}^3(B)$7 is a functor

$\mathrm{Cob}^3(B)$8

Its persistent Khovanov homology, with respect to a functor $\mathrm{Cob}^3(B)$9 to modules and homology $k\mathrm{Cob}^3(B)$0, is the composition

$k\mathrm{Cob}^3(B)$1

For any $k\mathrm{Cob}^3(B)$2 and any $k\mathrm{Cob}^3(B)$3, the $k\mathrm{Cob}^3(B)$4-persistent Khovanov homology is

$k\mathrm{Cob}^3(B)$5

with Betti polynomial

$k\mathrm{Cob}^3(B)$6

This is the fixed-boundary persistence theory described in the paper (Liu et al., 2024).

Without fixed boundary, a persistence tangle in $k\mathrm{Cob}^3(B)$7 is a functor

$k\mathrm{Cob}^3(B)$8

and its persistent Khovanov homology is

$k\mathrm{Cob}^3(B)$9

For $D$00,

$D$01

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 (Liu et al., 2024).

When $D$02, the direct sum

$D$03

has a natural $D$04-module structure via the shift map

$D$05

The same applies in the planar setting with $D$06. The paper remarks that under standard assumptions, the structure theorem and stability results for persistence modules apply, though details are not elaborated (Liu et al., 2024).

5. Functoriality, invariance, and induced maps

Bar-Natan invariance gives the starting point: the bracket complex is invariant up to chain homotopy in $D$07, and hence $D$08 is a tangle invariant up to chain homotopy. The functor

$D$09

maps isotopy classes of tangles to chain-homotopy classes (Liu et al., 2024).

For any additive module-valued functor

$D$10

one obtains

$D$11

and the resulting homology is an isotopy invariant. In particular, Theorem 3.2 states that

$D$12

maps isotopy classes of tangles to homotopy classes of cochain complexes (Liu et al., 2024).

The paper also gives explicit induced maps for generators of morphisms in $D$13. For the cap cobordism

$D$14

the induced map is

$D$15

hence

$D$16

For the cup cobordism

$D$17

one has

$D$18

hence

$D$19

For a local saddle $D$20, the map is described using the mapping-cone description via the crossing-change tangle $D$21 and the cochain map

$D$22

which induces

$D$23

These formulas enable step-by-step computation of persistent Khovanov homology along a filtration (Liu et al., 2024).

In the planar-algebra category $D$24, the paper constructs a cochain map

$D$25

for morphisms $D$26 induced by a $D$27-input planar tangle $D$28. The map acts on independent components as identity on arc $D$29 arc and circle $D$30 circle, while on arc $D$31 circle it is

$D$32

The chain-map property is verified by commuting diagrams with $D$33 and $D$34, and the construction

$D$35

is functorial. However, there is no notion of isotopy between tangles with different boundaries, so no isotopy invariance statement is claimed in $D$36 (Liu et al., 2024).

6. Computation, examples, applications, and limitations

The computational framework is the usual $D$37-state construction. Given a tangle diagram $D$38 with $D$39 crossings, one builds the cube of states $D$40, forms each smoothing $D$41, assembles the bracket complex

$D$42

shifts to

$D$43

applies $D$44 or $D$45, and computes homology with quantum grading

$D$46

For $D$47, each smoothing with $D$48 arcs and $D$49 circles contributes

$D$50

and each differential is a sum of local saddle maps with identity elsewhere (Liu et al., 2024).

The paper presents three small tangle examples. If $D$51 is a single left-handed crossing on an arc, the cochain complex collapses to

$D$52

Applying $D$53 gives

$D$54

Hence

$D$55

and

$D$56

If $D$57 is a single right-handed crossing on an arc, then

$D$58

and after applying $D$59,

$D$60

Hence

$D$61

with

$D$62

If $D$63 is a single arc with no crossings, then

$D$64

and

$D$65

These three tangles are equivalent up to Reidemeister moves, and their Khovanov homology groups and gradings of generators agree (Liu et al., 2024).

The applications emphasized in the paper are in KDA. For a planar tangle $D$66 and a center $D$67, one may define a single-center radial filtration by

$D$68

where $D$69 is the disk of radius $D$70 centered at $D$71. Then

$D$72

is a persistence tangle, and one computes

$D$73

to capture birth and death of local features around $D$74 as the window grows. For a finite collection of curves $D$75 and a generic projection $D$76 with only double crossings, the same construction using

$D$77

again defines a persistence tangle in $D$78 and yields persistent Khovanov homology of tangles as multiscale local descriptors (Liu et al., 2024).

Several limitations are explicitly identified. The approach inherits the $D$79 scaling of state-cube constructions in Khovanov theory, and explicit complexity bounds and specific data structures are not developed. In the varying-boundary category $D$80, 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 (Liu et al., 2024).

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 $D$81, $D$82, saddle, and $D$83. Its stated tradeoffs are more complex chain complexes, lack of an isotopy-invariance statement in $D$84, and growth of module sizes with the numbers of arcs and circles (Liu et al., 2024).