TopoOR: Topological & Orthogonal Frameworks

• TopoOR is a multifaceted framework that integrates topology optimization in functionally graded materials, multimodal operating room scene representation, and oriented matroid theory through topological and orthogonality principles.
• In topology optimization, TopoOR optimizes elastic moduli directly without intermediate densities, achieving superior convergence and efficiency, particularly in shear-dominated scenarios.
• Across applications, TopoOR maintains native geometric and combinatorial structures, resulting in enhanced expressivity, interpretability, and computational efficiency.

TopoOR refers to three distinct technical constructs in mathematical optimization, scene representation for surgical operating rooms, and oriented matroid theory, each rooted in the “topological” or “orthogonality” structure of the underlying domain. The term has been established independently across these fields, but consistently denotes frameworks that leverage higher-order relationships and algebraic topology for either modeling, optimization, or combinatorial analysis.

1. TopoOR in Functionally Graded Orthotropic Topology Optimization

In computational mechanics, TopoOR designates a topology optimization algorithm for linear orthotropic, functionally graded materials, as introduced by Aycart et al. (Ben-Yelun et al., 2023). This approach fundamentally generalizes standard SIMP (Solid Isotropic Material with Penalization) techniques by directly optimizing the set of six elastic moduli—three Young’s moduli ($E_1, E_2, E_3$) and three shear moduli ($G_{12}, G_{13}, G_{23}$)—per finite element, while holding Poisson ratios fixed but respecting symmetry and positive-definiteness constraints.

The global optimization objective is the joint homogenization (minimization of standard deviation) of the volumetric and shear strain-energy densities, establishing spatially graded orthotropic property fields. The method supports both a direct (strain-based) and complementary (stress-based) formulation, with the complementary version demonstrating superior convergence and efficiency, especially for shear-dominated problems such as torsion.

The update rule for each modulus in element $e$ at iteration $t$ is given as either:

• Strain-based: $$\alpha_e^{t+1} = \alpha_e^t [1 + (H_e - \bar{H}) / (s_H k)]$$
• Stress-based: $$\alpha_e^{t+1} = \alpha_e^t / [1 - (H_e - \bar{H}) / (s_H k)]$$

where $k$ is the step-size hyperparameter, $\alpha_e$ is the modulus, $H_e$ is the element-wise energy variable, and $s_H$ its standard deviation.

Key implementation features include direct operation on moduli (no intermediate densities), explicit enforcement of material constraints, and elimination of heuristic penalization. Application to various canonical loading scenarios exhibited significant gains versus isotropic or classical SIMP benchmarks, particularly in shear or mixed-mode loading.

2. TopoOR in Multimodal Operating Room Scene Representation

In surgical informatics, TopoOR refers to a unified topological scene representation for the operating room (Wang et al., 10 Mar 2026). The paradigm shifts from dyadic, graph-based surgical scene graphs (SSGs) to a “combinatorial complex” (CC) formalism, where polyadic interactions and heterogeneous modalities are encoded as higher-rank cells (0-cells: entities, 1-cells: interactions, 2-cells: group events).

Traditional SSGs are fundamentally limited by pairwise links and forced flattening of manifold-specific features into single-vector or token representations, thereby losing crucial spatial, kinematic, and temporal context. TopoOR augments expressive power by incorporating:

• Arbitrary-rank cells that support group interactions (e.g., events linking surgeon, robot, tool, and patient simultaneously),
• Multimodal preservation by separately mapping 3D geometry, audio, kinematic logs, and RGB features to their respective entities,
• Higher-Order Attention (HAT), which generalizes GAT by propagating messages across the entire incidence structure, including upward (boundary to co-boundary) and downward flows.

The layer-wise HAT update for cell $G_{12}, G_{13}, G_{23}$0 is:

$G_{12}, G_{13}, G_{23}$1

with attention scores incorporating rank-pair biases to preserve modality/type distinctions.

Empirical evaluation on the MM-OR dataset demonstrated that TopoOR matches competing methods for geometric tasks (sterility breach), but considerably outperforms all baselines for complex reasoning tasks (robot-phase prediction, next-action anticipation). Performance gains are linked to the ability of CCs to capture irreducible k-way constraints and maintain modality-specific information throughout the relational hierarchy.

Method	Sterility	Next Action	Robot Phase
MM2SG (LLM-based)	55.00	35.40	56.90
Vanilla Transf.	76.83	34.80	65.29
SurgLatentGraph	76.83	37.46	64.61
TopoOR (Ours)	76.83	41.10	73.53

These results suggest that combinatorial complex-based architectures, exemplified by TopoOR, subsume conventional scene-graph methods, achieving greater expressivity and enabling interpretable, safety-critical reasoning in a multimodal context.

3. Tope-Orthogonality Relations in Oriented Matroid Theory

The term “TopoOR” also denotes an orthogonality relation for decompositions of topes in the tope graph of a simple oriented matroid (Matveev, 2017). Here, a tope $G_{12}, G_{13}, G_{23}$2 is uniquely decomposed as a minimal signed sum of vertices belonging to a centrally symmetric cycle $G_{12}, G_{13}, G_{23}$3 in the tope graph. For two oriented matroids $G_{12}, G_{13}, G_{23}$4 on ground sets of sizes $G_{12}, G_{13}, G_{23}$5 (with $G_{12}, G_{13}, G_{23}$6), and for sufficiently large decomposition sets $G_{12}, G_{13}, G_{23}$7 of their respective topes, Matveev established that the associated “long” $G_{12}, G_{13}, G_{23}$8-vectors are orthogonal.

The construction proceeds as follows:

• For a given decomposition $G_{12}, G_{13}, G_{23}$9, form an abstract simplicial complex $e$0 whose facets are derived from the separation-sets between $e$1 and elements of $e$2.
• The “long” $e$3- and $e$4-vectors of these complexes, computed via explicit combinatorial rules, satisfy an orthogonality relation:

$e$5

This orthogonality emanates from Dehn–Sommerville-type palindromicity in the complexes, interpreted via eigenvectors for the backward identity involution.

A worked example illustrates the chain from tope decomposition, to facet and separation-set construction, to calculation of “long” $e$6-vectors, “long” $e$7-vectors, and final orthogonality verification.

4. Topological and Algebraic Structures Underlying TopoOR

In each context, TopoOR frameworks invoke constructs from algebraic topology and combinatorial theory:

• In optimization, the partitioning of energy into modal contributions (volumetric and shear), and the spatially varying manipulation of orthotropic moduli, reflect a rigid adherence to the underlying geometric structure of the material.
• In OR scene modeling, combinatorial complexes generalize simplicial complexes by allowing cells of arbitrary rank, and the associated incidence relations define the higher-dimensional neighborhood structure.
• In oriented matroids, representation via cycles and signed-sums explicitly mirrors facets and boundaries within the associated complex, while orthogonality arises from their algebraic symmetries.

Key to all variants is the avoidance of premature flattening: signals, moduli, or relational structures are maintained in their native geometric or combinatorial form, enabling richer modeling and inference.

5. Performance, Expressivity, and Computational Considerations

Across domains, TopoOR-powered architectures achieve notable gains in expressivity, interpretability, and (for the operating room representation) computational efficiency. For instance, in the functionally graded optimization setting, the separation of moduli optimizations bypasses volume or density penalization heuristics, yielding more efficient solution spaces and superior results in shear-dominated or mixed loading. In scene representation, TopoOR sustains, and often exceeds, the performance of purely graph- or LLM-based methods while using fewer parameters and with lower per-inference latency.

A plausible implication is that as systems become increasingly multi-modal and polyadic in their true relational structure, higher-order and “topologically aware” formalisms such as those implemented in TopoOR are likely to outperform pairwise or vector/tensor-flat models in both generalization and interpretability.

6. Limitations and Open Directions

Despite the increased expressivity, limitations remain. For TopoOR in the OR context:

• Current benchmarks focus on classification metrics; direct clinical impact remains unvalidated.
• Generalizing beyond rank-2 cells (i.e., to temporally extended interactions or continuous patient anatomy) is an active challenge.
• Few-shot adaptation and transfer across varying OR layouts are not yet addressed.

For the oriented matroid-theoretic TopoOR, the orthogonality relation applies under cardinality and parity constraints on the ground sets and decomposition sizes; more generalized forms or analogs in non-matroidal settings are unexplored.

Across all domains, further integration with differentiable topology, continuous complexes, or multi-physics settings represents a key direction for ongoing research.