[Paper Review] Essential meridional surfaces for tunnel number one knots

Mario Eudave-Muñoz|arXiv (Cornell University)|Aug 12, 1999

Geometric and Algebraic Topology7 references3 citations

TL;DR

This paper constructs infinitely many tunnel number one knots whose exteriors contain essential meridional surfaces of prescribed genus g and 2n boundary components, and also exhibits knots with n disjoint, non-parallel, closed incompressible surfaces each of genus n. The results demonstrate the rich and complex surface structure possible in tunnel number one knot complements, extending known geometric constraints in 3-manifold topology.

ABSTRACT

We show that for each pair of positive integers g and n, there are infinitely many tunnel number one knots, whose exteriors contain an essential meridional surface of genus g, and with 2n boundary components. We also show that for each positive integer n, there are tunnel number one knots whose exteriors contain n disjoint, non-parallel, closed incompressible surfaces, each of genus n.

Motivation & Objective

• To investigate the existence and structure of essential meridional surfaces in the exteriors of tunnel number one knots.
• To determine whether such knots can support essential surfaces with arbitrary genus and number of boundary components.
• To explore the possibility of embedding multiple disjoint, non-parallel, closed incompressible surfaces in the complement of a tunnel number one knot.
• To provide explicit constructions of such knots and surfaces, thereby expanding the known geometric complexity of tunnel number one knot complements.

Proposed method

• Using Dehn surgery and tangle replacement techniques, the author constructs specific families of tunnel number one knots.
• The construction ensures that the resulting knot exteriors contain embedded surfaces that are both essential and meridional.
• The genus and number of boundary components of the essential surfaces are controlled via combinatorial and topological parameters in the construction.
• The existence of multiple disjoint incompressible surfaces is established by ensuring geometric separation and incompressibility through careful embedding in the knot complement.
• The proof relies on standard tools in 3-manifold topology, including incompressibility, boundary incompressibility, and the classification of essential surfaces.
• The author uses the fact that tunnel number one knots admit a single unknotting tunnel, which is leveraged to control the topology of the complement and embed desired surfaces.

Experimental results

Research questions

• RQ1 Can tunnel number one knots have exteriors containing essential meridional surfaces of arbitrary genus g and 2n boundary components?
• RQ2 Are there tunnel number one knots whose exteriors contain multiple disjoint, non-parallel, closed incompressible surfaces?
• RQ3 What constraints, if any, limit the number and genus of such surfaces in tunnel number one knot complements?
• RQ4 How can explicit constructions of such knots and surfaces be achieved using geometric and topological methods?

Key findings

• For every pair of positive integers g and n, there exist infinitely many tunnel number one knots whose exteriors contain an essential meridional surface of genus g with 2n boundary components.
• For each positive integer n, there are tunnel number one knots whose exteriors contain n disjoint, non-parallel, closed incompressible surfaces, each of genus n.
• The constructed surfaces are proven to be essential, meaning they are incompressible, boundary-incompressible, and not boundary-parallel.
• The constructions demonstrate that tunnel number one knots can support highly non-trivial surface structures, defying simple geometric intuition.
• The results show that the topological complexity of tunnel number one knot complements is significantly richer than previously recognized.
• The paper provides a systematic method to generate such knots and surfaces, offering a new class of examples in geometric knot theory.