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Pushout Attachments and Conditional Complexity of Executable Models
Alexander Kolpakov
We represent the modular enlargement of an executable model by a pushout of finite typed presentations. Adhesivity preserves the original model and recovers the shared interface, w…
Random Knots via Stiefel manifolds
Alexander Kolpakov, Igor Rivin
A fixed simplex, randomly projected into three dimensions and joined in Hamiltonian order, produces a rich and unusually tractable model of random stick knots. We prove that Gaussi…
Discriminant Varieties for Stick Knots and Links
Alexander Kolpakov, Igor Rivin
How many knot types can be built from a fixed budget of straight sticks? We prove that the answer has factorial-scale growth, settling its order for the first time. No previously p…
DiRe-RAPIDS: Topology-faithful dimensionality reduction at scale
Alexander Kolpakov, Igor Rivin
Dimensionality reduction methods such as UMAP and t-SNE are central tools for visualising high-dimensional data, but their local-neighborhood objectives can preserve sampling noise…