topology

fkcompute: an efficient invariant calculator

arXiv:2607.12155

summary

The paper presents fkcompute, an open‑source tool for efficiently calculating the Gukov–Manolescu invariant of links given by braid presentations, and releases the first public database of these invariants.

Abstract

We introduce fkcompute, an open-source package for computing the Gukov--Manolescu invariant of links from a braid presentation. fkcompute implements Park's inverted state sum through a three-phase pipeline. First, a search is performed for a suitable braid presentation and for an additional inversion data on the braid. Then, the state space of the inverted sum is encoded as a polytope, bounded by the associated linear constraint system. Finally, the invariant is constructed by multiplication of R-matrices associated to the states. Benchmarks show that prime knots up to 12 crossings, and prime links up to 10 crossings and of at most 3 components, are comfortably within reach. As a result, fkcompute is used to compile the first public database of the Gukov--Manolescu invariant. The package is available as a Python library, a command-line tool, and a Mathematica paclet.

17 pages, 1 figure, 1 table, GitHub repository: github.com/caltech-quantum-topology/fkcompute, database: topology.fyi

Topics & keywords

#knot theory#link invariants#braid groups#computational topology#quantum invariantsGukov–Manolescu invariantinverted state sumR-matricespolytope encodingPython library
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