math.GT daily digest: 4 new submissions for 15 July 2026

New submissions

fkcompute: an efficient invariant calculator

• Authors: Paul Orland; Davide Passaro; Lara San Martín Suárez; Toby Saunders-A'Court; Josef Svoboda
• arXiv: arXiv:2607.12155
• Subjects: Geometric Topology (math.GT); Mathematical Software (cs.MS); Quantum Algebra (math.QA)
• Comments: 17 pages, 1 figure, 1 table, GitHub repository: this http URL, database: this http URL
• 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.

Genus-zero links with prescribed knots as components

• Authors: Raphael Appenzeller; José Pedro Quintanilha
• arXiv: arXiv:2607.12729
• Subjects: Geometric Topology (math.GT)
• Comments: 15 pages, 11 figures, comments welcome
• Abstract:

We prove that any finite collection of at least three isotopy classes of knots in a 3-manifold $M$ is realizable as the components of a genus-zero link in $M$, provided that an obvious requirement on their conjugacy classes in is met. This condition is vacuously satisfied for , and in this case we also control the pairwise linking numbers of the components. Replacing the 3-genus with the 4-genus, we obtain an analogous result where only two knot isotopy classes are prescribed.

Unknotting number, ribbon concordance, and singular instantons

• Authors: Hayato Imori; JungHwan Park; Masaki Taniguchi
• arXiv: arXiv:2607.12768
• Subjects: Geometric Topology (math.GT)
• Comments: 15 pages
• Abstract:

We use equivariant singular instanton Floer theory with the Chern--Simons filtration to obstruct same-sign unknotting operations. We show that, for a large class of slice knots obtained through ribbon concordance, any unknotting sequence of null-homologous twists must contain both signs. The same method gives a $3$--manifold analogue, obstructing certain homology $3$--spheres from surgery on a knot and providing evidence for the monotonicity of the Dehn surgery number under ribbon homology cobordism.

A Theorem of Wolpert, and some Variations

• Authors: Ludovico Battista; Nolwenn Le Quellec; Juan Souto
• arXiv: arXiv:2607.12977
• Subjects: Geometric Topology (math.GT)
• Comments: 15 pages + biblio. Comments are welcome!
• Abstract:

We give a streamlined proof of the fact that generically, isospectral hyperbolic surfaces are isometric. We also prove some versions of this result allowing for quasi-Fuchsian groups or considering the simple length spectrum.