Complexity of 3-manifolds obtained by Dehn filling

TL;DR

This work develops a systematic framework for bounding the complexity $c(M(\alpha))$ of closed 3-manifolds obtained by even Dehn fillings on a cusped manifold $M$ with torus boundary. Central to the approach is the slope-norm captured via the Farey tessellation and boundary-normal surface data, which yields a general lower bound and a constructive upper bound based on boundary layering and folding. The authors prove that for infinite families of even fillings there is a constant additive gap between the lower and upper bounds, with the gap depending only on a chosen triangulation (or its ideal version or a knot diagram), and provide explicit bounds such as $2k \le c(M(\alpha_k)) \le 2k + 13|\mathcal T| + 7$; they further extend this to ideal triangulations and knot exteriors, yielding bounds with mechanizable constants. The approach is demonstrated through concrete calculations for the figure eight knot complement, the pretzel knot $P(-2,3,7)$, and the trefoil, producing practical complexity estimates and illustrating the method's applicability using standard tools like Reginaregina and SnapPy. The results offer a practical toolkit for predicting complexity growth in Dehn-filled manifolds and enable controlled comparisons across large families of fillings.

Abstract

Let $M$ be a compact 3--manifold with boundary a single torus. We present upper and lower complexity bounds for closed 3--manifolds obtained as even Dehn fillings of $M.$ As an application, we characterise some infinite families of even Dehn fillings of $M$ for which our method determines the complexity of its members up to an additive constant. The constant only depends on the size of a chosen triangulation of $M$, and the isotopy class of its boundary. We then show that, given a triangulation $\mathcal T$ of $M$ with $2$--triangle torus boundary, there exist infinite families of even Dehn fillings of $M$ for which we can determine the complexity of the filled manifolds with a gap between upper and lower bound of at most $13 |\mathcal T| + 7.$ This result is bootstrapped to obtain the gap as a function of the size of an ideal triangulation of the interior of $M$, or the number of crossings of a knot diagram. We also show how to compute the gap for explicit families of fillings of knot complements in the three-sphere. The practicability of our approach is demonstrated by determining the complexity up to a gap of at most 10 for several infinite families of even fillings of the figure eight knot, the pretzel knot $P(-2,3,7)$, and the trefoil.

Figures (13)

• Figure 1: The Farey tessellation.
• Figure 2: Left: Torus boundary $\mathcal{T}_\partial$ of isotopy class $(a/b,c/d,(a+c)/(b+d)).$ The arrow indicates the folding over the diagonal, the dotted line indicates the target filling slope. Right: Corresponding ideal triangle(s) in the Farey tessellation. The arrow indicates source and target triangle, the bold vertex indicates the target filling slope.
• Figure 3: From $D$ to $P'.$ In the rightmost picture of $P'$, subdivisions are mostly omitted for readability. The blue lines denote the edges representing $K.$
• Figure 4: Removing a small neighbourhood of $L$ from a tetrahedron followed by triangulating the resulting truncated tetrahedron. Top row: $L$ meets the tetrahedron in an edge and a vertex. This results in a subdivision into $9+3=12$ tetrahedra. Bottom row: $L$ meets the tetrahedron in three vertices. This results in $13+3=16$ tetrahedra.
• Figure 5: Turning a degree one vertex (left), or a degree two vertex (second from left) into a vertex of degree $3$ with two or one flips respectively. Note that edges $e$ and $e'$ must be distinct because $\mathcal{T}_\partial$ is not a $2$-sphere.

Theorems & Definitions (15)

• Proposition 1: Corollary 10 in JRST21SlopeNorm
• Theorem 2: Corollary 2 in Jaco-norm-2020
• Corollary 3
• proof
• Remark 4
• Theorem 5
• proof
• Corollary 6
• proof
• Corollary 7