Constructing knots with low rational genera

2025/11/19 by Clayton McDonald, McDonald, Clayton, Allison N. Miller +1
Mathematics · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Advanced Operator Algebra Research

paper · pdf · doi:10.48550/arxiv.2511.15900

Abstract

We give a flexible construction for knots in the 3-sphere that bound surfaces of unexpectedly low genus in punctured open books on 3-manifolds. We use this construction to give the first examples of knots whose genus differs in different ℤ/2ℤ homology balls. We also establish that every knot bounds a Möbius band in a rational homology ball, and that there are knots whose genus in T⁴ and B⁴ differ arbitrarily.

Citations

• A note on rationally slice knots
• Relative genus bounds in indefinite four-manifolds
• Amphichiral knots with large 4-genus
• A calculation of Blanchfield pairings of 3-manifolds and knots
• Khovanov homology and the slice genus

Related

• Constructing Rational Homology 3-Spheres That Bound Rational Homology 4-Balls
• Rational genus and Heegaard Floer homology
• The computational complexity of knot genus in a fixed 3-manifold
• A 4-dimensional rational genus bound
• Heegaard Floer correction terms and rational genus bounds