Totally geodesic boundaries of knot complements

For each $g \ge 2$, we prove existence of a computable constant $\epsilon(g) > 0$ such that if $S$ is a strongly irreducible Heegaard surface of genus $g$ in a complete hyperbolic 3-manifold $M$ and $\gamma$ is a simple geodesic of length less than $\epsilon(g)$ in $M$, then $\gamma$ is isotopic into $S$.

Let $M$ be a compact hyperbolic $3$-manifold with volume $V$. Let $L$ be a link such that $M\setminus L$ is hyperbolic. For any hyperbolic link $L$ in $M$, in this article, we try to establish an upper bound of the length of $n^{th}$ shortest closed geodesic of $M\setminus L$ in terms of $V$.

This paper determines which orientable hyperbolic 3-manifolds contain simple closed geodesics. The Fuchsian group corresponding to the thrice-punctured sphere generates the only example of a complete non-elementary orientable hyperbolic 3-manifold that does not contain a simple closed geodesic. We do not assume that the manifold is geometrically finite or that it has finitely generated fundamental group.

We consider the existence of simple closed geodesics or "geodesic knots" in finite volume orientable hyperbolic 3-manifolds. Previous results show that at least one geodesic knot always exists [Bull. London Math. Soc. 31(1) (1999) 81-86], and that certain arithmetic manifolds contain infinitely many geodesic knots [J. Diff. Geom. 38 (1993) 545-558], [Experimental Mathematics 10(3) (2001) 419-436]. In this paper we show that all cusped orientable finite volume hyperbolic 3-man...

In this paper, we explicitly construct large classes of incommensurable hyperbolic knot complements with the same volume and the same initial (complex) length spectrum. Furthermore, we show that these knot complements are the only knot complements in their respective commensurabiltiy classes by analyzing their cusp shapes. The knot complements in each class differ by a topological cut-and-paste operation known as mutation. Ruberman has shown that mutations of hyperelliptic ...

We survey some tools and techniques for determining geometric properties of a link complement from a link diagram. In particular, we survey the tools used to estimate geometric invariants in terms of basic diagrammatic link invariants. We focus on determining when a link is hyperbolic, estimating its volume, and bounding its cusp shape and cusp area. We give sample applications and state some open questions and conjectures.

In this paper we prove that certain hyperbolic link complements of $2$-tori in $S^4$ do not contain closed embedded totally geodesic hyperbolic $3$-manifolds.

Suppose n>2, let M,M' be n-dimensional connected complete finite-volume hyperbolic manifolds with non-empty geodesic boundary, and suppose that the fundamental group of M is quasi-isometric to the fundamental group of M' (with respect to the word metric). Also suppose that if n=3, then the boundaries of M and of M' are compact. We show that M is commensurable with M'. Moreover, we show that there exist homotopically equivalent hyperbolic 3-manifolds with non-compact geodesic ...

In this article, we investigate the problem of counting totally geodesic surfaces in the complement of hyperbolic knots with at most 9 crossings. Adapting previous counting techniques of boundary slope and intersection, we establish uniqueness of a totally geodesic surface for the knots $7_4$ and $9_{35}$. Extending an obstruction to the existence of totally geodesic surfaces due to Calegari, we show that there is no totally geodesic surface in the complement of 47 knots.

Every closed geodesic $\gamma$ on a surface has a canonically associated knot $\widehat\gamma$ in the projective unit tangent bundle. We study, for $\gamma$ filling, the volume of the associated knot complement with respect to its unique complete hyperbolic metric. We provide a lower bound for the volume relative to the number of homotopy classes of $\gamma$-arcs in each pair of pants of a pants decomposition of the surface.