Constructing Rational Homology 3-Spheres That Bound Rational Homology 4-Balls

TL;DR

The paper advances the understanding of which rational surgeries yield 3-manifolds that bound rational homology balls by developing GOCL and IGOCL moves on negative definite plumbings and leveraging lattice-embedding criteria. It constructs three large families of new plumbed 3-manifolds bounding rational balls and provides explicit positive rational surgery coefficients for torus knots that realize these bounds, uniting complementary-leg theory, Lisca's graphs, and branched-cover techniques. A key methodological contribution is the Complicated Method, based on chi-slice ribbon surfaces in equivariant Kirby diagrams, which certifies when a constructed graph bounds a rational ball. The results extend known classifications (including GALL and Lecuona) and yield new instances supporting slice-ribbon-type phenomena in arborescent and Montesinos knot families, with potential implications for a broader understanding of rational homology cobordisms and their boundaries.

Abstract

We present three large families of new examples of plumbed 3-manifolds that bound rational homology 4-balls. These are constructed using two operations, also defined here, that preserve the lack of a lattice embedding obstruction to bounding rational homology balls. Apart from in the cases shown in this paper, it remains open whether these operations are rational homology cobordisms in general. The families of new examples include a multitude of families of rational surgeries on torus knots, and we explicitly describe which positive torus knots we now know to have a surgery that bounds a rational homology ball. While not the focus of this paper, we implicitly confirm the slice-ribbon conjecture for new, more complicated, examples of arborescent knots, including many Montesinos knots.

Figures (35)

• Figure 1: These are the graphs obtainable by performing IGOCL and GOCL moves on the linear graph $(-3,-2,-3,-3,-3)$. Here the length of the chain of $-2$'s is $k\geq 0$, $(a_1,\dots,a_{m_1})$ and $(\alpha_1,\dots,\alpha_{m_2})$ are complementary sequences, and $(b_1,\dots,b_{n_1})$ and $(\beta_1,\dots,\beta_{n_2})$ are complementary sequences.
• Figure 2: The form of all graphs obtainable from $(-3,-2,-2,-3)$ using GOCL moves. Here the sequences $(a_1,\dots,a_{l_1})$ and $(\alpha_1,\dots,\alpha_{l_2})$ are complementary, as well as the sequences $(b_1,\dots, b_{m_1})$ and $(\beta_1,\dots,\beta_{m_2})$, and the sequences $(z_1,\dots, z_{n_1})$ and $(\zeta_1,\dots,\zeta_{n_2})$.
• Figure 3: The form of all graphs obtainable from $(-2,-2,-3,-4)$ by GOCL and IGOCL moves. Here the length of the chain of $-2$'s is $k\geq 0$, $(a_1,\dots,a_{m_1})$ and $(\alpha_1,\dots,\alpha_{m_2})$ are complementary sequences, and $(b_1,\dots,b_{n_1})$ and $(\beta_1,\dots,\beta_{n_2})$ are complementary sequences.
• Figure 4: Kirby diagram associated to a plumbing.
• Figure 5: Example of a Riemenschneider diagram representing the complementary fractions $[5,3,2,2]^-$ and $[2,2,2,3,4]^-$.

Theorems & Definitions (36)

• Proposition 1: Corollary of Donaldson's Theorem
• Theorem A
• Remark
• Proposition 2: The Simple Method
• Theorem B
• Remark 6.1
• Definition 7
• Definition 8
• Definition 9
• Definition 10