Polynomial bounds for surfaces in cusped 3-manifolds

TL;DR

This work resolves the longstanding question of bounding isotopy classes of embedded essential surfaces in cusped 3-manifolds by establishing explicit polynomial bounds that apply to broad classes including classical alternating links, virtual alternating knots, toroidally alternating links, and many Dehn fillings. The authors develop and leverage a generalized chunk decomposition, normal form theory, and combinatorial area to control subsurfaces and their boundary patterns, producing universal bounds that depend only on diagram crossing number and surface Euler characteristic, via a bounding parameter X for incompressible subsurfaces in ambient chunks. The results generalize prior polynomial bounds for closed and spanning surfaces to all embedded essential surfaces (orientable or nonorientable, with boundary of any slope), with explicit formulas and effective constants, and extend to Dehn-filled manifolds and virtual torus settings through a unified framework. The methods offer a novel approach beyond triangulation-based normal surface theory, providing practical, computable bounds and shedding light on the prevalence of quasifuchsian surfaces and related phenomena in cusped hyperbolic 3-manifolds, as well as implications for Dehn-filling behavior and broader knot families.

Abstract

It is natural to ask how many isotopy classes of embedded essential surfaces lie in a given 3-manifold. The first bounds on the number of such surfaces were exponential, using normal surfaces. More recently, by restricting to alternating link complements in 3-sphere, Hass, Thompson and Tsvietkova obtained polynomial bounds, but for a limited class of surfaces: closed and spanning ones. Here, we complete the picture for classical alternating links and extend these results to other classes of cusped 3-manifolds. We give explicit polynomial bounds on all embedded essential surfaces, closed or any boundary slope, orientable or non-orientable. Our 3-manifolds are complements of links with alternating diagrams on wide classes of surfaces in broad families of 3-manifolds. This includes all alternating links in 3-sphere as well as many non-alternating ones, alternating virtual knots, many toroidally alternating knots, and most Dehn fillings of such manifolds.

Figures (3)

• Figure 1: Left: a surface with meridianal boundary can be isotoped to meet $N(L)$ transversely away from crossings. Right: in the chunk decomposition, $\partial Z$ cuts off exactly two corners of truncation faces.
• Figure 2: Left: The disk $D$ meeting distinct curves on $T^2\times\{0\}$. Right: The disk $E$ and the disk $D\cup E$.
• Figure 3: Left: an arc of $\gamma$ cannot run to the same side of $D$. Right: Thus $F\cap (T^2\times[0,\epsilon_1]$ is the union of an annulus and a Möbius band, for $\epsilon_1$ small.

Theorems & Definitions (50)

• Lemma 2.7: Lemma 8.5 of Paper1
• Theorem 2.8: Theorem 11.4 of Paper1
• Lemma 3.1
• proof
• Lemma 3.2
• Lemma 4.1
• Remark 4.2
• proof : Proof of Lemma \ref{['Lem:BddSurfaceCount']}
• Lemma 4.3
• proof