Constructing knots with low rational genera

TL;DR

The authors develop a flexible, Cobordism-based construction that, from a 3-manifold $M$ and a knot $J\subset M$, produces links in $S^3$ bounding low-genus surfaces in punctured open books on $M$. By lifting appropriate tangle cobordisms in $M^\circ\times I$ to surfaces in $Spin(M)^\circ$ and using Heegaard-diagram techniques, they obtain knots with small genus in rational homology balls such as $Spin(M)^\circ$ while exploiting Casson-Gordon signatures to obstruct smaller genera in ordinary 4-balls. They prove notable results: every knot bounds a Möbius band in $Spin(\mathbb{RP}^3)^\circ$; there exist knots with genus two in $Spin(L(3,1))^\circ$ but with $g_4$ forcing larger values, and they construct arbitrarily large gaps between $g_4$ and genus bounds in ambient manifolds, including $(T^4)^\circ$. The techniques combine tangle cobordisms, diagrammatic realizations, and Casson-Gordon invariants to produce and obstruct low-genus slicings, advancing understanding of how genus behaves across rational and integral homology contexts.

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 $\mathbb{Z}/2\mathbb{Z}$ homology balls. We also establish that every knot bounds a M{ö}bius band in a rational homology ball, and that there are knots whose genus in $T^4$ and $B^4$ differ arbitrarily.

Figures (14)

• Figure 1: The knots $K(J_0,J_1,J_2,J_3,J_4)$ we use to prove Theorem \ref{['thm:mainthm']}, subject to certain conditions on the companion knots $J_0,J_1,J_2,J_3,J_4$.
• Figure 2: In black, we have $M^\circ\times [-1,1]$, with the extra 4-dimensional puncture in red as $B^3 \times [-1/2,0]$. In blue, we have the surface $F_{[-1,0]} \cup F_0 \cup F_{[0,1]}$.
• Figure 3: Left: The initial tangle $T_J$. Center: Four curves appear. Right: Canceling half twists appear, together with four orange bands that will be added in the next step.
• Figure 4: The tangle obtained by attaching bands (left), after isotopy (center), and after yet another isotopy (right).
• Figure 5: (Left) In a neighborhood of the Heegaard torus, we see $J$ together with $k$ oppositely oriented pairs of $\alpha$- and $\beta$-curves. (Center) We have introduced a canceling pair of half full twists via isotopy. (Right) We have further introduced a braid $\sigma$ and its inverse $\sigma^{-1}$.

Theorems & Definitions (41)

• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Remark 1.6
• Theorem 1.7
• Theorem 1.8
• Proposition 2.2
• proof
• Remark 2.3
• Remark 2.4