[Paper Review] Construction of spines of two-bridge link complements and upper bounds of their Matveev complexities

Masaharu Ishikawa, Keisuke Nemoto|arXiv (Cornell University)|Mar 12, 2015

Geometric and Algebraic Topology10 references3 citations

TL;DR

This paper constructs explicit almost-simple spines for two-bridge link complements in the 3-sphere, providing tight upper bounds on their Matveev complexities. By leveraging continued fraction representations of two-bridge links and analyzing pillowcase spines, the authors derive a precise upper bound formula and determine the exact complexity for an infinite family of links of the form $[2,1,\ldots,1,2]$, proving it equals $2n-2$. The method also yields improved upper bounds for meridian-cyclic branched coverings of these links.

ABSTRACT

We give upper bounds of the Matveev complexities of two-bridge link complements by constructing their spines explicitly. In particular, we determine the complexities for an infinite sequence of two-bridge links corresponding to the continued fractions of the form [2,1,...,1,2]. We also give upper bounds for the 3-manifolds obtained as meridian-cyclic branched coverings of the 3-sphere along two-bridge links.

Motivation & Objective

• To establish explicit upper bounds on the Matveev complexity of two-bridge link complements using constructive spine theory.
• To determine the exact complexity for an infinite sequence of two-bridge links with continued fraction $[2,1,\ldots,1,2]$.
• To improve existing upper bounds on the complexity of meridian-cyclic branched coverings of two-bridge links.
• To provide a systematic method for constructing spines of two-bridge link complements based on continued fraction decomposition.

Proposed method

• The authors construct almost-simple spines for two-bridge link complements by decomposing the link into tangles and building pillowcase spines for each tangle segment.
• Each tangle with $a_i$ twists is represented by a spine $A_i$ — a sphere with four holes and an internal disk — called a pillowcase, which collapses the tangle to a polyhedron.
• The full spine of the link complement is formed by gluing these pillowcases along their boundary components, preserving the almost-simple structure.
• The number of true vertices in the resulting spine is computed as $\sum_{i=1}^{n}a_i + 2(n-3) - \sharp\{a_i=1\}$, which serves as an upper bound for the complexity.
• For the infinite family $[2,1,\ldots,1,2]$, the bound simplifies to $2n-2$, and this value is shown to be exact via volume and ideal triangulation arguments.
• The method is extended to branched coverings by lifting the spine and adding meridian disks, with correction terms depending on whether $p$ is odd or even.

Experimental results

Research questions

• RQ1 What is the Matveev complexity of two-bridge link complements with continued fraction $[2,1,\ldots,1,2]$?
• RQ2 Can an explicit spine construction yield tighter upper bounds on Matveev complexity than existing methods for two-bridge links?
• RQ3 How does the complexity of meridian-cyclic branched coverings of two-bridge links scale with the covering degree $d$?
• RQ4 To what extent do the constructed spines achieve the minimal number of ideal tetrahedra in ideal triangulations of two-bridge link complements?
• RQ5 Is the upper bound derived from the spine construction sharp for hyperbolic two-bridge links?

Key findings

• The Matveev complexity of a two-bridge link complement with continued fraction $[a_1,\ldots,a_n]$, where $a_i > 0$ and $a_1,a_n > 1$, is bounded above by $\sum_{i=1}^{n}a_i + 2(n-3) - \sharp\{a_i=1\}$.
• For the infinite family of links with continued fraction $[2,1,\ldots,1,2]$ of length $n$, the exact complexity is $2n-2$, which matches the upper bound.
• The upper bound for the complexity of the $d$-fold meridian-cyclic branched covering of a two-bridge link is $d\left(\sum_{i=1}^{n}a_i + 2(n-3) - \sharp\{a_i=1\}\right) + rd$, where $r=1$ if $p$ is odd and $r=3$ if $p$ is even.
• The constructed spines yield topological ideal triangulations with at most the number of ideal tetrahedra equal to the number of true vertices, and this bound is tighter than previous results from canonical decompositions.
• For the $[2,1,\ldots,1,2]$ family, the upper bound from this construction matches the bound from Sakuma and Weeks’ canonical triangulation, confirming optimality in this case.
• The method improves upon earlier upper bounds for branched coverings, as shown in comparison with Petronio and Vesnin’s results.