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Kazuhiro Ichihara

Publications and source records attributed to Kazuhiro Ichihara.

At least 73 records · Page 4Linked to original sources
arXiv2012

On the maximal number and the diameter of exceptional surgery slope sets

Concerning the set of exceptional surgery slopes for a hyperbolic knot, Lackenby and Meyerhoff proved that the maximal cardinality is 10 and the maximal diameter is 8. Their proof is computer-aided in part, and both bounds are achieved simultaneously. In this note, it is observed that the diameter bound 8 implies the maximal cardinality bound 10 for exceptional surgery slope sets. This follows from the next known fact: For a hyperbolic knot, there exists a slope on the peripheral torus such that all exceptional surgery slopes have distance at most two from the slope. We also show that, in generic cases, that particular slope above can be taken as the slope represented by the shortest geodesic on a horotorus in a hyperbolic knot complement.

math.GT↗
arXiv2011

Exceptional surgeries on $(-2,p,p)$-pretzel knots

We give a complete description of exceptional surgeries on pretzel knots of type $(-2, p, p)$ with $p \ge 5$. It is known that such a knot admits a unique toroidal surgery yielding a toroidal manifold with a unique incompressible torus. By cutting along the torus, we obtain two connected components, one of which is a twisted $I$-bundle over the Klein bottle. We show that the other is homeomorphic to the one obtained by certain Dehn filling on the magic manifold. On the other hand, we show that all such pretzel knots admit no Seifert fibered surgeries.

math.GT↗
arXiv2010

Constructions of surface bundles with rank two fundamental groups

We give a construction of hyperbolic 3-manifolds with rank two fundamental groups and report an experimental search to find such manifolds. Our manifolds are all surface bundles over the circle with genus two surface fiber. For the manifolds so obtained, we then examine whether they are of Heegaard genus two or not. As a byproduct, we give an infinite family of fibered knots of genus two in the 3-sphere whose knot groups are of rank two.

math.GT↗
arXiv2010

Gromov hyperbolicity and a variation of the Gordian complex

We introduce new simplicial complexes by using various invariants and local moves for knots, which give generalizations of the Gordian complex defined by Hirasawa and Uchida. In particular, we focus on the simplicial complex defined by using the Alexander-Conway polynomial and the Delta-move, and show that the simplicial complex is Gromov hyperbolic and quasi-isometric to the real line.

math.GT↗
arXiv2008

Surgical distance between lens spaces

It is well-known that any pair of closed orientable 3-manifolds are related by a finite sequence of Dehn surgeries on knots. Furthermore Kawauchi showed that such knots can be taken to be hyperbolic. In this article, we consider the minimal length of such sequences connecting a pair of 3-manifolds, in particular, a pair of lens spaces.

math.GT↗
arXiv2007

Strongly invertible knots, rational-fold branched coverings and hyperbolic spatial graphs

A construction of a spatial graph from a strongly invertible knot was developed by the second author, and a necessary and sufficient condition for the given spatial graph to be hyperbolic was provided as well. The condition is improved in this paper. This enable us to show that certain classes of knots can yield hyperbolic spatial graphs via the construction.

math.GN↗
arXiv2007

Lens spaces obtainable by surgery on doubly primitive knots

In this paper, we consider which lens spaces are obtainable by Dehn surgery described by Berge on doubly primitive knots. It is given an algorithm to decide whether a given lens space is obtainable by such surgery. Also included is a complete characterization of such surgery yielding lens spaces with Klein bottles.

math.GT↗
arXiv2007

Lower bounds on boundary slope diameters for Montesinos knots

In this paper, two lower bounds on the diameters of the boundary slope sets are given for Montesinos knots. One is described in terms of the minimal crossing numbers of the knots, and the other is related to the Euler characteristics of essential surfaces with the maximal/minimal boundary slopes.

math.GT↗
arXiv2007

Crosscap numbers of pretzel knots

The crosscap number of a knot in the 3-sphere is defined as the minimal first Betti number of non-orientable subsurfaces bounded by the knot. In this paper, we determine the crosscap numbers of pretzel knots. The key ingredient to obtain the result is the algorithm of enumerating all essential surfaces for Montesinos knots developed by Hatcher and Oertel.

math.GT↗
arXiv2005

Integral non-hyperbolike surgeries

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

math.GT↗