🤖 AI Summary
This work addresses the problem of efficiently computing the Gukov–Manolescu invariant from braid representations by proposing a three-stage approach: searching for suitable braids and reversal data, encoding the state space as a polyhedron under linear constraints, and constructing the invariant via products of R-matrices. The method enables, for the first time, scalable computation of this invariant and yields the first publicly available database. Integrating polyhedral encoding, systems of linear constraints, R-matrix operations, and optimized search algorithms, the framework supports computations for prime knots up to 12 crossings and prime links with up to 10 crossings and at most three components. Accompanying software includes a Python library, a command-line tool, and a Mathematica package.
📝 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.