The Non-Orientable Four-Ball Genus of a New Infinite Family of Torus Knots

2025/07/16 by Sinha, Shreya
#Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2507.12606

Abstract

We extend previous work by using a combination of band surgeries and known bounds to compute γ₄(T_{4n, (2n±1)² + 4n-2}) = 2n-1 for all n ≥ 1. We further generalize this result by showing that γ₄(T_{4n + 2k, n(4n + 2k) - 1}) = γ₄(T_{4n + 2k, (n+2)(4n + 2k) - 1}) = 2n-1 + k for all n ≥ 1 and k ≥ 0. All knots in this family are counterexamples to Batson's conjecture.

Citations

• Non-orientable 4-genus of torus knots
• The Non-Orientable 4-Genus of 11 Crossing Non-Alternating Knots
• On the nonorientable 4-genus of double twist knots
• On the nonorientable four-ball genus of torus knots
• The non-orientable 4-genus for knots with 10 crossings
• An infinite family of counterexamples to Batson's conjecture
• Nonorientable surfaces bounded by knots: a geography problem
• A counterexample to Batson's conjecture

Related

• An infinite family of counterexamples to Batson's conjecture
• On the nonorientable four-ball genus of torus knots
• Nonorientable four-ball genus can be arbitrarily large
• Non-orientable 4-genus of torus knots
• The Non-Orientable 4-Genus of 11 Crossing Non-Alternating Knots