Discriminant Varieties for Stick Knots and Links

Abstract

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 published general upper bound improves on the exponential-in-the-square estimate obtained from crossing-number enumeration; we replace it with a factorial-scale upper bound, which is optimal at the level of growth order. The proof turns polygonal self-intersection into a sparse real-algebraic chamber problem in only linearly many dimensions, while a complementary braid construction supplies factorially many distinct knots. The result creates a direct bridge between knot topology, real algebraic geometry, fewnomial structure, and permutation combinatorics.

Disclosure

“Acknowledgments The authors used OpenAI’s GPT-5.6-sol as a research and coding assistant to help strengthen the results, improve the exposition, and generate and audit the Python and Lean 4 code accompanying this paper. The authors reviewed the resulting mathematics and code and take resp”

PDF page 15

Classification

Rewriting existing author-written text

Multiplier

4

Verified

2026-08-23T19:19:27.390241Z

Structural counts