Kazuhiro Ichihara

Accidental surfaces in knot complements

It is well known that for many knot classes in the 3-sphere, every closed incompressible surface in their complements contains an essential loop which is isotopic into the boundary of the knot exterior. In this paper, we investigate closed incompressible surfaces in knot complements with this property. We show that if a closed, incompressible, non-boundary-parallel surface in a knot complement has such loops, then they determine the unique slope on the boundary of the knot exterior. Moreover, if the slope is non-meridional, then such loops are mutually isotopic in the surface. As an application, a necessary and sufficient condition for knots to bound totally knotted Seifert surfaces is given.

Verified Computations for Hyperbolic 3-Manifolds

For a given cusped 3-manifold M admitting an ideal triangulation, we describe a method to rigorously prove that either M or a filling of M admits a complete hyperbolic structure via verified computer calculations. Central to our method is an implementation of interval arithmetic and Krawczyk’s test. These techniques represent an improvement over existing algorithms as they are faster while accounting for error accumulation in a more direct and user-friendly way.

Cyclic and finite surgeries on Montesinos knots

We give a complete classification of the Dehn surgeries on Montesinos knots which yield manifolds with cyclic or finite fundamental groups.

Boundary slopes of non-orientable Seifert surfaces for knots

We study non-orientable Seifert surfaces for knots in the 3-sphere, and examine their boundary slopes. In particular, it is shown that for a crosscap number two knot, there are at most two slopes which can be the boundary slope of its minimal genus non-orientable Seifert surface, and an infinite family of knots with two such slopes will be described. Also, we discuss the existence of essential non-orientable Seifert surfaces for knots.

Alexander polynomials of doubly primitive knots

We give a formula for Alexander polynomials of doubly primitive knots. This also gives a practical algorithm to determine the genus of any doubly primitive knot.

HYPERBOLIC KNOT COMPLEMENTS WITHOUT CLOSED EMBEDDED TOTALLY GEODESIC SURFACES

It is conjectured that a hyperbolic knot complement does not contain a closed embedded totally geodesic surface. In this paper, we show that there are no such surfaces in the complements of hyperbolic 3-bridge knots and double torus knots. Some topological criteria for a closed essential surface failing to be totally geodesic are given. Roughly speaking, sufficiently ‘complicated’ surfaces cannot be totally geodesic.

KNOTS WITH ARBITRARILY HIGH DISTANCE BRIDGE DECOMPOSITIONS

We show that for any given closed orientable 3-manifold M with a Heegaard surface of genus g, any positive integers b and n, there exists a knot K in M which admits a (g,b)-bridge splitting of distance greater than n with respect to the Heegaard surface except for (g,b) = (0,1), (0,2).

Non-left-orderable surgeries and generalized Baumslag–Solitar relators

We show that a knot has a non-left-orderable surgery if the knot group admits a generalized Baumslag–Solitar relator and satisfies certain conditions on a longitude of the knot. As an application, it is shown that certain positively twisted torus knots admit non-left-orderable surgeries.

INTEGRAL NON-HYPERBOLIKE SURGERIES

It is shown that a hyperbolic knot in the 3-sphere admits at most nine integral surgeries yielding non-hyperbolike 3-manifolds; namely, 3-manifolds which are reducible or whose fundamental groups are not infinite word-hyperbolic.

Accidental surfaces and exceptional surgeries

, which is dened as follows. Let be a closed essential (i.e., incom-pressible and not @ -parallel) surface in a knot exterior. We call accidental if it con-tains a non-trivial loop which is isotopic into the peripher al torus of the knot. Thereare some motivations to study the accidental surface from th e topological or the geo-metrical viewpoint. For example, it is known that accidenta l surfaces in a hyperbolicknot complement have a particular geometric behavior [11]. See [7] for more detail.First, we will consider the accidental slope for accidental surfaces. A slope on theperipheral torus of a knot is determined by the isotopy from a non-trivial loop on anaccidental surface into the torus. It is shown in [7, Theorem 1] that the slope is inde-pendent of the choice of the non-trivial loop on the surface. Hence we call this slopethe accidental slope for the accidental surface. In contrast, the accidental slo pe is notdetermined uniquely for a knot. In fact, an example of a knot a dmitting two accidentalsurfaces with accidental slopes 0 and 1 was given in [7, Figure 1]. We know that anyaccidental slope is integral or meridional [1, Lemma 2.5.3] , and the example showsthat there is a knot with integral and meridional accidental slopes. Hence, it is naturalto ask how many integral accidental slopes exist for a knot. In this paper, we give abound of the minimal intersection number of accidental slop es.Theorem 3.2. Let