Research

The braid axis of a closed 3-braid lifts to a genus one fibered knot in the double cover of S^3 branched over the closed braid.  Every genus one fibered knot in a 3-manifold may be obtained in this way.  Using this perspective we answer a question of Morimoto about the number of genus one fibered knots in lens spaces.  We determine the number of genus one fibered knots up to homeomorphism in any given lens space.  This number is 3 in the case of the lens space L(4,1), 2 for the lens spaces L(m,1) with m>0, and at most 1 otherwise.

We construct an algorithm that lists all closed essential surfaces in the complement of a knot that lies on the fiber of a trefoil or figure eight knot. Such knots are Berge knots and hence admit lens space surgeries. Furthermore they may have arbitrarily large hyperbolic volume. Using this algorithm we concoct large volume Berge knots of two flavors: those whose complement contains arbitrarily many distinct closed essential surfaces, and those whose complement contains no closed essential surfaces.

Using Kirby Calculus, we explicitly pass from Berge's R-R descriptions of ten families of knots with lens space surgeries to surgery descriptions on the minimally twisted five chain link (MT5C). Since the MT5C admits a strong involution, we also give the corresponding tangle descriptions.

By obtaining surgery descriptions of knots which lie on the genus one fiber of the trefoil or figure eight knot, we show that these include hyperbolic knots with arbitrarily large volume. These knots admit lens space surgeries and form two families of Berge knots. By way of tangle descriptions we also obtain surgery descriptions for these knots on minimally twisted chain links.

In a lens space X of order r a knot K representing an element of the fundamental group pi_1 X = Z/rZ of order s \leq r contains an orientable surface S properly embedded in its exterior X-N(K) such that \bdry S intersects the meridian of K minimally s times.  Assume S has just one boundary component.  Let g be the minimal genus of such surfaces for K, and assume s \geq 4g-1.  Then with respect to the genus one Heegaard splitting of X, K has bridge number at most 1.

We study knots that lie as essential simple closed curves on the fiber of a genus one fibered knot in S^3.  We determine certain surgery descriptions of these knots that enable estimates on volumes of these knots.  We also develop an algorithm to list all closed essential surfaces in the complement of a given knot in this family.  Relationships between the volumes of such knots and the surfaces in their exteriors is then examined.

This article is the offspring of [2] and [5].  The main idea is to examine what lens spaces may be obtained by surgery on large volume hyperbolic knots in lens spaces.  It probably should be retitled as the current title suggests something more grand.

Since all knots in lens spaces are rationally null-homologous, they each have a properly embedded orientable once punctured surface in their exterior whose boundary may cross the meridian of the knot more than once.  We call such surfaces rational Seifert surfaces.  A projection of knot in a lens space onto the Heegaard torus gives rise to a diagram for the knot.  From this diagram we will show how to construct a rational Seifert surface.