Multi-Topological Representations

• Multi-topological representations are formal frameworks that encode multiple interacting topological structures across scales, labels, and morphisms.
• They integrate methodologies such as multiscale persistence-diagram embeddings, multi-parameter persistent homology, and simplicial deep learning to capture complex data features.
• Practical applications include enhanced medical image classification, efficient graph drawing enumeration, and computable representations in measure theory and quiver systems.

Multi-topological representations are formalisms that encode several interacting topological structures within a single mathematical or computational object. In current usage, the term covers at least eight distinct but related constructions: multiscale Hilbert embeddings of persistence diagrams; multi-parameter and multi-species persistent homology; multi-scale and multi-filtration topological encoders for medical images; simplicial neural architectures fused with persistent-homology descriptors; quiver-shaped systems of spaces with homology and homotopy invariants; Type-2 multi-representations of measurable sets in computable measure spaces; exhaustive generation of topological graph drawings under beyond-planarity constraints; and diagrammatic data for higher-dimensional topological orders with loop- and membrane-like excitations (Reininghaus et al., 2014, Natarajan et al., 3 Mar 2026, Gu et al., 8 Dec 2025, Verma et al., 2024, Li et al., 2020, Wu, 2010, Angelini et al., 2019, Huang et al., 2024). Taken together, these works suggest that the common principle is not a single invariant, but the simultaneous organization of topology across multiple scales, labels, morphisms, or excitation types.

1. Multiscale persistence-diagram embeddings

A central line of work represents topology through persistence diagrams and then lifts those diagrams into spaces compatible with statistical learning. A persistence diagram $D$ is the multiset of points $(b,d)\in\mathbb{R}^2$, $d>b$, recording the birth and death times of homological features in a filtration. Comparison is classically mediated by the $p$-Wasserstein distance

$$W_p(D_1,D_2)=\biggl(\inf_\gamma \sum_{u\in D_1}\|u-\gamma(u)\|_\infty^p\biggr)^{1/p},$$

with the diagonal added with infinite multiplicity, and in particular

$$W_1(D_1,D_2)=\inf_\gamma \sum_{u\in D_1}\|u-\gamma(u)\|_\infty.$$

The difficulty is that $W_1$ does not embed in a Hilbert space, which motivates a kernel construction based on an $L_2$-valued feature map (Reininghaus et al., 2014).

For each scale $\sigma>0$, the feature map

$$\Phi_\sigma:\text{Diagrams}\to L_2(\Omega),\qquad \Omega=\{x=(x_1,x_2):x_2\ge x_1\},$$

is defined as the solution at time $(b,d)\in\mathbb{R}^2$0 of a heat-diffusion PDE with Dirichlet boundary condition on the diagonal. In closed form,

$(b,d)\in\mathbb{R}^2$1

where $(b,d)\in\mathbb{R}^2$2 is the reflection of $(b,d)\in\mathbb{R}^2$3 across the diagonal. The associated kernel is

$(b,d)\in\mathbb{R}^2$4

A genuinely multiscale kernel is then formed by a finite sum

$(b,d)\in\mathbb{R}^2$5

This construction is positive definite because $(b,d)\in\mathbb{R}^2$6 lies in the Hilbert space $(b,d)\in\mathbb{R}^2$7, so $(b,d)\in\mathbb{R}^2$8 is an inner product; any finite sum of such kernels remains positive definite. It is also stable with respect to the $(b,d)\in\mathbb{R}^2$9-Wasserstein distance: $d>b$0 The section-level significance is that the representation converts persistent topology into a kernel geometry compatible with kernel SVMs and kernel PCA while retaining a Lipschitz control under diagram perturbations.

The closed-form evaluation of $d>b$1 for diagrams of sizes $d>b$2 and $d>b$3 requires $d>b$4 time. In the reported applications, the numbers of persistence points in the $d>b$5- and $d>b$6-dimensional diagrams are typically in the low hundreds, so direct evaluation is efficient. Empirically, the kernel was used with C-SVMs on SHREC 2014 3D shape classification and retrieval and on Outex_TC_00000 texture recognition. On synthetic SHREC, $d>b$7 achieved up to $d>b$8 accuracy versus $d>b$9 for the persistence-landscape kernel $p$0; on real SHREC, up to $p$1 versus $p$2; and on textures it improved over $p$3 by $p$4–$p$5, with $p$6 versus $p$7 for CLBP-S and $p$8 versus $p$9 for CLBP-M. The same account emphasizes that summing over several $$W_p(D_1,D_2)=\biggl(\inf_\gamma \sum_{u\in D_1}\|u-\gamma(u)\|_\infty^p\biggr)^{1/p},$$0 yields a multiresolution signature interpolating between fine topology and coarse topology.

2. Multi-parameter and multi-species persistent homology

A second meaning of multi-topological representation arises when topology is indexed simultaneously by spatial scale and by subsets of labels or species. Let $$W_p(D_1,D_2)=\biggl(\inf_\gamma \sum_{u\in D_1}\|u-\gamma(u)\|_\infty^p\biggr)^{1/p},$$1 be labeled point sets in $$W_p(D_1,D_2)=\biggl(\inf_\gamma \sum_{u\in D_1}\|u-\gamma(u)\|_\infty^p\biggr)^{1/p},$$2, and for $$W_p(D_1,D_2)=\biggl(\inf_\gamma \sum_{u\in D_1}\|u-\gamma(u)\|_\infty^p\biggr)^{1/p},$$3 write $$W_p(D_1,D_2)=\biggl(\inf_\gamma \sum_{u\in D_1}\|u-\gamma(u)\|_\infty^p\biggr)^{1/p},$$4. At scale $$W_p(D_1,D_2)=\biggl(\inf_\gamma \sum_{u\in D_1}\|u-\gamma(u)\|_\infty^p\biggr)^{1/p},$$5, one forms a Čech complex

$$W_p(D_1,D_2)=\biggl(\inf_\gamma \sum_{u\in D_1}\|u-\gamma(u)\|_\infty^p\biggr)^{1/p},$$6

or a Vietoris–Rips complex

$$W_p(D_1,D_2)=\biggl(\inf_\gamma \sum_{u\in D_1}\|u-\gamma(u)\|_\infty^p\biggr)^{1/p},$$7

In practice, especially in $$W_p(D_1,D_2)=\biggl(\inf_\gamma \sum_{u\in D_1}\|u-\gamma(u)\|_\infty^p\biggr)^{1/p},$$8D and $$W_p(D_1,D_2)=\biggl(\inf_\gamma \sum_{u\in D_1}\|u-\gamma(u)\|_\infty^p\biggr)^{1/p},$$9D, the chromatic Delaunay–Čech complex is used because it has the same homotopy type but far fewer simplices. The indexing poset is

$$W_1(D_1,D_2)=\inf_\gamma \sum_{u\in D_1}\|u-\gamma(u)\|_\infty.$$0

ordered by $$W_1(D_1,D_2)=\inf_\gamma \sum_{u\in D_1}\|u-\gamma(u)\|_\infty.$$1 if $$W_1(D_1,D_2)=\inf_\gamma \sum_{u\in D_1}\|u-\gamma(u)\|_\infty.$$2 and $$W_1(D_1,D_2)=\inf_\gamma \sum_{u\in D_1}\|u-\gamma(u)\|_\infty.$$3. The assignment $$W_1(D_1,D_2)=\inf_\gamma \sum_{u\in D_1}\|u-\gamma(u)\|_\infty.$$4 is then a multi-parameter filtration, and the resulting persistent homology is the functor $$W_1(D_1,D_2)=\inf_\gamma \sum_{u\in D_1}\|u-\gamma(u)\|_\infty.$$5 sending $$W_1(D_1,D_2)=\inf_\gamma \sum_{u\in D_1}\|u-\gamma(u)\|_\infty.$$6 to $$W_1(D_1,D_2)=\inf_\gamma \sum_{u\in D_1}\|u-\gamma(u)\|_\infty.$$7 (Natarajan et al., 3 Mar 2026).

The same framework classifies four types of topological features. For an inclusion $$W_1(D_1,D_2)=\inf_\gamma \sum_{u\in D_1}\|u-\gamma(u)\|_\infty.$$8, with induced map $$W_1(D_1,D_2)=\inf_\gamma \sum_{u\in D_1}\|u-\gamma(u)\|_\infty.$$9, a class may be common to multiple species, may be present in some species but disappear in the presence of others, may only become visible when multiple species are considered together, or may be formed by some species and remain visible in the presence of others. Algebraically, these correspond to intersections of images, to nonzero elements of $W_1$0, to elements of $W_1$1, and to nonzero classes in $W_1$2. Operationally, each map $W_1$3 produces four persistence diagrams: domain, kernel, cokernel, and image.

The algorithmic pipeline restricts to subsets $W_1$4 of size at most $W_1$5, with $W_1$6 or $W_1$7, builds chromatic Delaunay–Čech filtrations on $W_1$8, computes ordinary persistent homology in dimensions $W_1$9 and $L_2$0, then computes persistent kernel, cokernel, and image for each inclusion $L_2$1 using matrix-reduction methods such as clearing and lock-free implementations. Each persistence diagram is vectorized by summary statistics—births, deaths, lifetimes, entropy, and counts—into a $L_2$2–$L_2$3-dimensional vector, followed by concatenation and optional PCA. The filtration is stored using simplex indices with color bitsets and boundary lists; the persistent-homology library cited is lophat, and persistence-of-maps routines are cited through phimaker. In the planar case, Delaunay triangulation per $L_2$4 is $L_2$5, chromatic Čech is $L_2$6, and although matrix reduction is roughly $L_2$7, sparse/clearing methods are near-linear in practice. The combinatorial burden grows like $L_2$8, which remains manageable for $L_2$9 and $\sigma>0$0.

Two applications are reported. In a synthetic tumor micro-environment model with $\sigma>0$1 cell types and $\sigma>0$2 parameter pairs, $\sigma>0$3 labeled point clouds were summarized into spatial-signature vectors with $\sigma>0$4 dimensions per single, $\sigma>0$5 per pair, and $\sigma>0$6 per triple, totaling $\sigma>0$7 features. $\sigma>0$8-means with $\sigma>0$9 recovered clusters aligned with the “elimination,” “equilibrium,” and “escape” regimes. In colorectal cancer tissue samples, $$\Phi_\sigma:\text{Diagrams}\to L_2(\Omega),\qquad \Omega=\{x=(x_1,x_2):x_2\ge x_1\},$$0 ROIs from $$\Phi_\sigma:\text{Diagrams}\to L_2(\Omega),\qquad \Omega=\{x=(x_1,x_2):x_2\ge x_1\},$$1 patients and $$\Phi_\sigma:\text{Diagrams}\to L_2(\Omega),\qquad \Omega=\{x=(x_1,x_2):x_2\ge x_1\},$$2 immune plus $$\Phi_\sigma:\text{Diagrams}\to L_2(\Omega),\qquad \Omega=\{x=(x_1,x_2):x_2\ge x_1\},$$3 stromal markers yielded pairwise and triple interactions across $$\Phi_\sigma:\text{Diagrams}\to L_2(\Omega),\qquad \Omega=\{x=(x_1,x_2):x_2\ge x_1\},$$4 combinations; a Random-Forest classifier with $$\Phi_\sigma:\text{Diagrams}\to L_2(\Omega),\qquad \Omega=\{x=(x_1,x_2):x_2\ge x_1\},$$5-fold CV reached mean balanced accuracy $$\Phi_\sigma:\text{Diagrams}\to L_2(\Omega),\qquad \Omega=\{x=(x_1,x_2):x_2\ge x_1\},$$6 for adenoma versus carcinoma ROIs, and feature importance highlighted periostin–macrophage pairwise interactions and macrophage–periostin–neutrophil and macrophage–periostin–SMA triples. A plausible implication is that multi-topological representations become informative precisely when interactions are not reducible to single-label geometry.

3. Learned multi-topological representations in vision and simplicial deep learning

In medical image classification, multi-topological representation has been formulated as the joint extraction of multi-scale and multi-filtration cubical persistent homology, followed by neural encoding and fusion with conventional visual backbones. For a $$\Phi_\sigma:\text{Diagrams}\to L_2(\Omega),\qquad \Omega=\{x=(x_1,x_2):x_2\ge x_1\},$$7D image $$\Phi_\sigma:\text{Diagrams}\to L_2(\Omega),\qquad \Omega=\{x=(x_1,x_2):x_2\ge x_1\},$$8 on a rectangular grid $$\Phi_\sigma:\text{Diagrams}\to L_2(\Omega),\qquad \Omega=\{x=(x_1,x_2):x_2\ge x_1\},$$9, the grid is viewed as a cubical complex, and a filtration function $(b,d)\in\mathbb{R}^2$00 induces the super-level filtration $(b,d)\in\mathbb{R}^2$01. The framework downsamples each image to $(b,d)\in\mathbb{R}^2$02 resolutions,

$(b,d)\in\mathbb{R}^2$03

computes cubical persistence diagrams in dimensions $(b,d)\in\mathbb{R}^2$04 and $(b,d)\in\mathbb{R}^2$05 at each scale, and then uses a “vineyard” algorithm to track features across adjacent scales through Hungarian matching, retaining only vines with matching threshold $(b,d)\in\mathbb{R}^2$06 and stability threshold $(b,d)\in\mathbb{R}^2$07. The stability score is

$(b,d)\in\mathbb{R}^2$08

The two filtrations are raw intensity $(b,d)\in\mathbb{R}^2$09 and gradient magnitude $(b,d)\in\mathbb{R}^2$10, producing four stable diagrams across $(b,d)\in\mathbb{R}^2$11 and $(b,d)\in\mathbb{R}^2$12. These are processed by a PointNet-STN-style encoder with cross-attention among filtrations, generating a topological embedding $(b,d)\in\mathbb{R}^2$13 that is injected into multiple CNN or Transformer layers through a topology-conditioned channel gate (Gu et al., 8 Dec 2025).

The gate takes the form

$(b,d)\in\mathbb{R}^2$14

and modulates features by

$(b,d)\in\mathbb{R}^2$15

Training uses the combined loss

$(b,d)\in\mathbb{R}^2$16

Across ISIC 2018, Kvasir, and CBIS-DDSM, and across ResNet152, SENet154, and SwinV2-B, the method reports consistent gains over strong baselines. On ISIC 2018, for example, ResNet152 rises from $(b,d)\in\mathbb{R}^2$17 to $(b,d)\in\mathbb{R}^2$18 with PHG-Net topology, to $(b,d)\in\mathbb{R}^2$19 with multi-scale vineyard under a single filtration, and to $(b,d)\in\mathbb{R}^2$20 with multi-filtration. On CBIS-DDSM with SwinV2-B, Euclidean/Wasserstein point-distance yields $(b,d)\in\mathbb{R}^2$21, persistence-scaled $(b,d)\in\mathbb{R}^2$22, and the paper’s relative-persistence metric $(b,d)\in\mathbb{R}^2$23. The interpretive claim made in the work is that $(b,d)\in\mathbb{R}^2$24 features capture isolated regions and $(b,d)\in\mathbb{R}^2$25 features capture cavities or ring-like structures, while multi-scale vines surviving the tracking procedure correspond to anatomical structures spanning resolutions.

A broader neural formulation is given by TopNets, which unifies simplicial message passing with persistent-homology vectorization. At layer $(b,d)\in\mathbb{R}^2$26, each simplex $(b,d)\in\mathbb{R}^2$27 carries features $(b,d)\in\mathbb{R}^2$28, receives boundary and coboundary messages, and is updated by

$(b,d)\in\mathbb{R}^2$29

Persistent homology is computed from a learnable filtration $(b,d)\in\mathbb{R}^2$30, vectorized by a pointwise embedding $(b,d)\in\mathbb{R}^2$31 and weight $(b,d)\in\mathbb{R}^2$32, and attached to simplices through

$(b,d)\in\mathbb{R}^2$33

These topological vectors are fused with simplicial message-passing features by a dimension-wise topological aggregator, then pooled across layers and simplex dimensions. The framework subsumes or generalizes methods such as RePHINE and TOGL, extends to continuous dynamics through a Graph-ODE formulation, and admits $(b,d)\in\mathbb{R}^2$34-equivariant geometric filtrations based on invariant quantities such as simplex volume or maximum pairwise distance (Verma et al., 2024).

The theoretical statement is that persistent-homology descriptors can provably enhance the expressivity of simplicial message-passing networks: Proposition 3.1 asserts the existence of non-isomorphic clique complexes $(b,d)\in\mathbb{R}^2$35 that simplicial Weisfeiler–Leman cannot distinguish, while color-based persistent homology yields different diagrams. Empirically, TopNets are reported on graph classification, molecular property regression, antibody CDR-H3 co-design, and molecular dynamics. On the SAbDab CDR-H3 benchmark, adding PH to a TransformerConv base improves amino-acid recovery from $(b,d)\in\mathbb{R}^2$36 to $(b,d)\in\mathbb{R}^2$37, while continuous ODE variants typically improve by $(b,d)\in\mathbb{R}^2$38–$(b,d)\in\mathbb{R}^2$39 points on classification and regression. Higher-order PH in dimensions $(b,d)\in\mathbb{R}^2$40 further improves IMDB-B and PROTEINS, with a $(b,d)\in\mathbb{R}^2$41–$(b,d)\in\mathbb{R}^2$42 GPU-time overhead relative to vanilla message passing and $(b,d)\in\mathbb{R}^2$43 memory in depth for continuous ODEs.

4. Quiver-shaped systems and computable multi-representations

Outside machine learning, multi-topological representation also denotes a system of spaces indexed by a quiver. For a finite quiver $(b,d)\in\mathbb{R}^2$44, a topological representation is a family

$(b,d)\in\mathbb{R}^2$45

with each $(b,d)\in\mathbb{R}^2$46 a topological space and each $(b,d)\in\mathbb{R}^2$47 continuous. Morphisms are vertex-wise continuous maps commuting with every arrow square, giving the category $(b,d)\in\mathbb{R}^2$48. This category is equivalent to a full subcategory $(b,d)\in\mathbb{R}^2$49-$(b,d)\in\mathbb{R}^2$50 of left $(b,d)\in\mathbb{R}^2$51-topological systems built from the path semigroup with zero, and it stands in parallel to the classical equivalence between $(b,d)\in\mathbb{R}^2$52-linear quiver representations and $(b,d)\in\mathbb{R}^2$53-modules. The same paper defines $(b,d)\in\mathbb{R}^2$54-limits, homology groups

$(b,d)\in\mathbb{R}^2$55

homotopy of morphisms in $(b,d)\in\mathbb{R}^2$56, and the assembling functor

$(b,d)\in\mathbb{R}^2$57

where $(b,d)\in\mathbb{R}^2$58 along arrows (Li et al., 2020).

The $(b,d)\in\mathbb{R}^2$59-limit is the inverse-limit construction applied degreewise after the singular-chain functor, and it makes homology sensitive to combinatorial structure in the quiver. The paper proves a Parallel Homotopy Axiom: if two morphisms are homotopic in $(b,d)\in\mathbb{R}^2$60, then they induce the same maps on $(b,d)\in\mathbb{R}^2$61. It also defines homotopy groups by $(b,d)\in\mathbb{R}^2$62, with $(b,d)\in\mathbb{R}^2$63 preserving homotopy equivalence between morphisms. For finite connected quivers, positive grading of every object in $(b,d)\in\mathbb{R}^2$64-$(b,d)\in\mathbb{R}^2$65, the symmetry of every cycle, the existence of an arrow-positive degree function, and vertex-positive grading are equivalent conditions. The examples show that a linear quiver with identity maps recovers the homology and homotopy of a single space, that a cycle quiver with rotations on $(b,d)\in\mathbb{R}^2$66 yields parity-dependent $(b,d)\in\mathbb{R}^2$67, and that convex affine gluings are homotopy equivalent to a point.

In computable analysis, the phrase multi-representation has a different technical sense: a partial surjection $(b,d)\in\mathbb{R}^2$68. Within a computable measure space $(b,d)\in\mathbb{R}^2$69, two such multi-representations are studied for measurable sets. The first, $(b,d)\in\mathbb{R}^2$70, is based on a convergence relation $(b,d)\in\mathbb{R}^2$71 that distinguishes finite-measure and infinite-measure behavior and is topologically complete among representations continuous with respect to $(b,d)\in\mathbb{R}^2$72 for which $(b,d)\in\mathbb{R}^2$73 is open. With respect to $(b,d)\in\mathbb{R}^2$74, $(b,d)\in\mathbb{R}^2$75 and $(b,d)\in\mathbb{R}^2$76 are decidable, $(b,d)\in\mathbb{R}^2$77 is computable, union is computable on all of $(b,d)\in\mathbb{R}^2$78, while intersection, difference, and complement are computable only on the domains explicitly identified in the theorem. The second, $(b,d)\in\mathbb{R}^2$79, replaces $(b,d)\in\mathbb{R}^2$80 by the induced probability measure

$(b,d)\in\mathbb{R}^2$81

where $(b,d)\in\mathbb{R}^2$82 comes from an effectively constructed partition of $(b,d)\in\mathbb{R}^2$83 in the infinite-measure case. This representation is admissible, hence topologically complete, for $(b,d)\in\mathbb{R}^2$84-continuous representations, makes all standard set-operations $(b,d)\in\mathbb{R}^2$85 computable on the entire $(b,d)\in\mathbb{R}^2$86, and is recursively complete in the class $(b,d)\in\mathbb{R}^2$87 (Wu, 2010).

These two literatures use the same word for structurally different objects. A plausible implication is that “multi-topological representation” is best understood as a family resemblance: in one case, a quiver-indexed topology of spaces and maps; in the other, a name-based representation theory for measurable sets whose completeness is measured by continuity and computability.

5. Enumerating topological realizations of graphs

In graph drawing, topological representation refers to a simple topological drawing of a graph on the sphere: vertices are distinct points and edges are Jordan arcs, with no self-crossings, no crossings between adjacent edges, and at most one crossing per pair of edges. The problem addressed in beyond-planarity is to generate all non-isomorphic topological representations of complete or complete bipartite graphs subject to local crossing constraints such as $(b,d)\in\mathbb{R}^2$88-planarity, $(b,d)\in\mathbb{R}^2$89-quasiplanarity, fan-planarity, fan-crossing-freeness, and $(b,d)\in\mathbb{R}^2$90-gap-planarity. Angelini et al. give a systematic generation procedure up to homeomorphism, using planarizations and recursive vertex insertion (Angelini et al., 2019).

The algorithm is “add-one-vertex.” Starting from a graph $(b,d)\in\mathbb{R}^2$91, one maintains a set of planarizations of all non-isomorphic drawings in the chosen class $(b,d)\in\mathbb{R}^2$92. For a fixed planarization $(b,d)\in\mathbb{R}^2$93, one first inserts the edge $(b,d)\in\mathbb{R}^2$94 by enumerating valid half-pathways in the dual graph: a half-pathway starts in a face incident to $(b,d)\in\mathbb{R}^2$95, ends in a destination face, and is constrained by a prohibited-edge set preventing self-crossings and class violations. After placing $(b,d)\in\mathbb{R}^2$96, the remaining edges $(b,d)\in\mathbb{R}^2$97 are routed by valid pathways from the face containing $(b,d)\in\mathbb{R}^2$98 to faces incident to each neighbor. Candidate outputs are filtered by an isomorphism test preserving cyclic edge orders around vertices, face incidence, and crossing orders along edges.

The paper gives correctness through three statements. Termination follows because every half-pathway has length at most the number of edges in the planarization, so every branch either inserts an edge or backtracks finitely. Completeness follows because every simple $(b,d)\in\mathbb{R}^2$99-drawing of $d>b$00 can be obtained by removing the highest-labeled vertex and retracing the dual paths of its incident edges. Soundness follows because the recursion never permits an insertion violating simplicity or the local crossing rules of $d>b$01. The worst-case time and space are exponential in $d>b$02, although the authors emphasize that prohibited-edge pruning and isomorphism elimination reduce the practical search substantially.

The reported low-order enumerations recover tight extremal bounds. For $d>b$03-planarity, $d>b$04 has $d>b$05 drawings, $d>b$06 has $d>b$07, $d>b$08 has $d>b$09, and $d>b$10 has none, so $d>b$11 is $d>b$12-planar iff $d>b$13. For $d>b$14-planarity, $d>b$15 has $d>b$16 non-isomorphic drawings and $d>b$17 has none, so $d>b$18 is $d>b$19-planar iff $d>b$20. For $d>b$21-planarity, $d>b$22 has $d>b$23 drawings and $d>b$24 has none, so $d>b$25 is $d>b$26-planar iff $d>b$27. Similar certificate/nonexistence results are given for $d>b$28, $d>b$29, $d>b$30, and the negative cases $d>b$31, $d>b$32, $d>b$33, and $d>b$34. Here the multi-topological aspect lies in the exhaustive comparison of distinct topological realizations of the same combinatorial graph under different local crossing axioms.

6. Diagrammatic data for higher-dimensional topological orders

In $d>b$35 dimensions and above, topological orders support extended excitations such as loops and membranes, so a representation of the order must encode not only fusion but also shrinking processes. The diagrammatic formalism introduced for $d>b$36D and $d>b$37D topological orders treats elementary fusion diagrams as vectors in fusion spaces $d>b$38, shrinking diagrams as vectors in shrinking spaces $d>b$39, and, in $d>b$40D, hierarchical-shrinking diagrams as vectors associated with successive shrinkings. Thick, medium, and thin lines distinguish excitations in the top level set $d>b$41, the first-shrink image $d>b$42, and the second-shrink image $d>b$43 (Huang et al., 2024).

If $d>b$44 fuse to $d>b$45 with $d>b$46 channels, the corresponding trivalent fusion diagram gives orthonormal basis vectors $d>b$47 for $d>b$48. Shrinking $d>b$49 to $d>b$50 with $d>b$51 channels gives basis vectors $d>b$52 for $d>b$53. In $d>b$54D, two-step shrinking $d>b$55 yields basis vectors for

$d>b$56

Basis changes between different decompositions are encoded by unitary $d>b$57-, $d>b$58-, and $d>b$59-symbols. $d>b$60-moves are fusion associators, $d>b$61-moves compare “shrink then fuse” versus “fuse then shrink” in $d>b$62D, and $d>b$63-moves do the analogous job for hierarchical shrinking in $d>b$64D.

Consistency imposes polynomial equations. The $d>b$65-symbols satisfy a pentagon equation, exactly as in ordinary anyon theory. The interaction of shrinking and fusion in $d>b$66D yields a shrinking-fusion hexagon equation; one numerical consequence is

$d>b$67

In $d>b$68D, a hierarchical hexagon involving $d>b$69, $d>b$70, and $d>b$71 compares three ways of relating $d>b$72, $d>b$73, and $d>b$74, together with a corresponding numerical consistency relation involving sums over intermediate shrinking and fusion labels. The paper conjectures that all anomaly-free higher-dimensional topological orders must satisfy these pentagon and hexagon constraints, and that violations indicate a quantum anomaly.

This framework places multi-topological representation at a categorical and diagrammatic extreme. Instead of encoding a single homotopy or homology object, it organizes fusion, shrinking, and hierarchical shrinking as a coherent algebra of vector spaces and unitary transformations. A plausible implication is that the same structural theme seen in multiscale persistent homology—compatibility across several topological views—reappears here as compatibility across several topological processes.