2022/08/16 by Jim Hoste, Hoste, Jim, Patrick D. Shanahan +3 · 2 citations
Computer Science · Materials Science · Mathematics · #57K10 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Supramolecular Self-Assembly in Materials #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2208.07850
openalex publication_date 2022/08/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate the nonorientable 4-genus γ4 of a special family of 2-bridge knots, the twist knots and double twist knots C(m,n). Because the nonorientable 4-genus is bounded by the nonorientable 3-genus, it is known that γ4(C(m,n)) ≤ 3. By using explicit constructions to obtain upper bounds on γ4 and known obstructions derived from Donaldson's diagonalization theorem to obtain lower bounds on γ4, we produce infinite subfamilies of C(m,n) where γ4=0,1,2, and 3, respectively. However, there remain infinitely many double twist knots where our work only shows that γ4 lies in one of the sets \1,2\, \2,3\, or \1,2,3\. We tabulate our results for all C(m,n) with |m| and |n| up to 50. We also provide an infinite number of examples which answer a conjecture of Murakami and Yasuhara.