Temporal Knot Module Overview

• Temporal Knot Module is a mathematical framework that captures the evolution of knot, link, or tangle features using persistent Khovanov homology.
• It integrates filtered chain complexes and functorial planar algebra constructions to track topological changes across a continuous time parameter.
• The approach computes barcode invariants representing birth–death intervals, offering a detailed temporal mapping from local to global topological structure.

A Temporal Knot Module is a mathematical construct for analyzing the emergence and persistence of topological features in knots, links, or tangles as they evolve over a real-valued parameter, interpreted as “time.” Built within the framework of persistent Khovanov homology, it combines filtered chain complexes, concrete module-valued functors via planar algebra, and the computation of birth–death intervals (“barcodes”) for topological features. This approach provides a multifaceted, functorial, and temporally resolved viewpoint on the local and global structure of curve-type data, revealing when and where knotting phenomena arise in space–time (Liu et al., 2024).

1. Classical Khovanov Chain Complex for Tangles

Let $T$ be an oriented tangle with $n$ crossings, $n_+$ positive and $n_-$ negative. The construction proceeds via the cube of resolutions: each vertex $s\in\{0,1\}^n$ encodes a smoothing $T_s$, yielding arcs and circles. Two gradings are assigned:

• Homological grading: $i(s) = \ell(s) - n_-$, with $\ell(s)$ the sum of smoothing choices.
• Quantum grading: Each circle is labeled from $V=R\{v_+, v_-\}$, $\deg(v_+)=1$, $\deg(v_-)=−1$.

The bigraded chain complex is

$$C^{i,j}(T) = \bigoplus_{s:\,i(s)=i} R\langle \text{labelings of circles in } T_s \rangle^{j-\theta}$$

with differential $d: C^{i,j}(T) \rightarrow C^{i+1,j}(T)$ induced by saddle cobordism. The chain homology $H^{i,j}(T) = H^i(C^{*,j}(T), d)$ recovers the Khovanov homology of $T$.

2. Filtration via Real-Parameter and Filtered Complexes

The temporally resolved variant is established by endowing the tangle $T$ with a real “time” parameter $\alpha \in \mathbb{R}$ via a filtration $T_\alpha \subset T_\beta$ for $\alpha \le \beta$. A practical case is $T_\alpha = T \cap D_\alpha$, where $D_\alpha$ is a Euclidean disk of radius $\alpha$.

For each $\alpha$,

$$C^{i,j}_\alpha(T) = \bigoplus_{k \le \alpha} C^{i,j}(T_k) \subseteq C^{i,j}(T)$$

with restricted differential $$d_\alpha^{i,j}: C^{i,j}_\alpha(T) \to C^{i+1,j}_\alpha(T)$$. This leads to a filtered complex whose homology

$H^{i,j}_\alpha(T) = H^i(C^{*, j}_\alpha(T))$

forms a persistence module indexed by $\alpha$.

If $\alpha \le \beta$, subcomplex inclusions $$i_{\alpha \rightarrow \beta}: C^{*,*}_\alpha(T) \hookrightarrow C^{*,*}_\beta(T)$$ produce induced maps at the homology level, constructing a functorial system $\{H^{i,j}_\alpha(T)\}_{\alpha \in \mathbb{R}}$.

3. Functorial Planar Algebraic Module Construction

To give the theory concrete computational form, Liu–Shen–Wei employ a functor

$$\Phi: \mathcal{P}la \longrightarrow \mathsf{Ch}^\bullet(\mathsf{Mod}_R)$$

from a planar algebraic category of tangles to chain complexes of $R$–modules.

For $T$ with $r(T)$ circles and $t(T)$ arcs,

$$\Phi(T) = V^{\otimes r(T)} \otimes W^{\otimes t(T)}, \quad V=R\{v_+, v_-\},\; W=R\{w\},\; \deg w = -1$$

Explicit $R$-linear maps for local cobordisms (such as saddle, cap, cup) instantiate the Khovanov cube differential in the module category, satisfying module-level versions of Bar-Natan’s relations.

Composing with the classical Khovanov bracket functor, this yields

$$\Phi \circ Kh: \mathcal{P}la \longrightarrow \mathsf{Ch}^\bullet(\mathsf{Mod}_R)$$

whose homology $H^{i,j}_\alpha(T; \Phi \circ Kh)$ supplies the bigraded persistence modules, with planar algebra morphisms directly encoding the inclusion maps for temporal filtration.

4. Persistence Invariants: Barcodes and Temporal Features

The principal invariants derived from Temporal Knot Modules are barcodes—collections of intervals in parameter $\alpha$ reflecting the lifespan of homological features.

Decomposition over a field $R=\Bbbk$ is given by

$$H^{i,j}(T) \cong \bigoplus_k\; \Bbbk[z]/(z^{b_k} - 0) \oplus \bigoplus_\ell\; \Bbbk[z]$$

yielding intervals $[\text{birth}_k,\, \text{death}_k)$ or $[\text{birth}_\ell,\, \infty)$ for generators in each bigrading.

• Short bars: Local features (e.g., loops, crossings with fleeting existence)
• Long bars ($+\infty$): Global features, persistent throughout “time”
• Quantum grading $j$: Encodes Jones-like charge of each feature

This barcode encodes the emergence and vanishing of knotting features as the tangle evolves in time, providing a multiscale, temporally resolved topological fingerprint.

5. Worked Example: The Single-Crossing Arc

Consider $T=$ an arc with a single under-crossing. The two smoothings yield arcs at heights $-1$ and $0$, giving

$$0 \rightarrow C^{-1,*}(T) = R\{w\} \xrightarrow{d} C^{0,*}(T) = R\{w \otimes v_+, w \otimes v_-\} \rightarrow 0$$

with $d(w) = w \otimes v_-$. Only $\Phi(w \otimes v_+)$ at homological degree $0$, quantum degree $0$, is nontrivial.

Suppose the arc appears at radius $\alpha=0$: then for $\alpha < 0$, the complex vanishes, and for $\alpha \ge 0$, $C^{*,*}_\alpha(T) = C^{*,*}(T)$. This produces a barcode for $H^{0,0}_\alpha(T)$: $$H^{0,0}_\alpha(T) = \begin{cases} 0, & \alpha < 0 \ R, & \alpha \ge 0 \end{cases}$$ yielding the interval $[0, \infty)$ for the generator, precisely recording the birth of the under-crossing in time.

6. Significance and Applications

The Temporal Knot Module construction determines the local “times” at which knotting features in curve-type data arise and vanish. It provides a robust framework for knot data analysis, leveraging persistent homology for localized, multiscale feature detection in data science applications involving spatially or temporally structured curves.

The functorial approach via planar algebra ensures boundary independence and full compatibility with topological changes, while barcode invariants allow detailed characterization and comparison across datasets and disciplines.

7. Summary of Temporal Knot Module Structure

A Temporal Knot Module for tangle $T$ consists of:

• The family of bigraded persistent modules $$\{H^{i,j}_\alpha(T; \Phi \circ Kh)\}_{\alpha \in \mathbb{R}}$$
• The functorial system of planar algebra embeddings encoding temporal inclusion
• Filtered chain complexes parameterized by real “time”
• Explicit computation and interpretation of birth–death intervals (barcodes) for every topological feature

This structure enables precise temporal mapping of knot-theoretic data, with each interval in the barcode revealing the spatial and temporal locus of knotting phenomena as encoded by persistent Khovanov homology (Liu et al., 2024).