It is experimentally known that achiral hyperbolic 3-manifolds are quite sporadic at least among those with small volume, while we can find plenty of them as amphicheiral knot complements in the 3-sphere. In this paper, we show that there exist in...

to appear in Osaka Journal of Mathematics [Refereed]

We present various examples of cosmetic bandings on knots and links, that is, bandings on knots and links leaving their types unchanged. As a byproduct, we give a hyperbolic knot which admits exotic chirally cosmetic surgeries yielding hyperbolic ...

Tetsuya Abe, In Dae Jong, John Luecke, John Osoinach

International Mathematics Research Notices Feb 2015 [Refereed]

We prove that for any integer n there exist infinitely many different knots in S3 such that n-surgery on those knots yields the same 3-manifold. In particular, when |n|=1 homology spheres arise from these surgeries. This answers Problem 3.6(D) on ...

Proceedings of the japan Academy, Series A, Mathematical Sciences 90(3) 54-56 Mar 2014 [Refereed]

We give a complete classification of toroidal Seifert fibered surgeries on alternating knots. Precisely, we show that if an alternating knot KK admits a toroidal Seifert fibered surgery, then KK is either the trefoil knot and the surgery slope is ...

On positive knots of genus two

In Dae Jong, Kengo Kishimoto

Kobe Journal of Mathematics 30(1--2) 1-18 Nov 2013 [Refereed]

We show that positive knots of genus two are positive-alternating or almost
positive-alternating. We also show that positive knots of genus two are quasialternating.
In addition, we show that every prime positive knot of genus two is obtained
from...

Tetsuya Abe, In Dae Jong, Yuka Omae, Masanori Takeuchi

Mathematical Proceedings of the Cambridge Philosophical Society 155(3) 219-235 Sep 2013 [Refereed]

We give a method for obtaining infinitely many framed knots which represent a diffeomorphic 4-manifold. We also study a relationship between the n-shake genus and the 4-ball genus of a knot. Furthermore we give a construction of homotopy 4-spheres...

Contemporary Mathematics 597(2013) 321-336 Aug 2013 [Refereed]

We give a new criterion for a given knot to be a Montesinos knot by using the Rasmussen invariant and the signature. We apply the criterion to study Seifert fibered surgery on a strongly invertible knot, and show that a (p, q, q)-pretzel knot with...

Topology and its Applications 159(4) 1064-1073 Mar 2012 [Refereed]

We give a new criterion for a given knot to be a Montesinos knot by using the Rasmussen invariant and the signature. We apply the criterion to study Seifert fibered surgery on a strongly invertible knot, and show that a (p, q, q)-pretzel knot with...

Topology and its Applications 159(4) 1007-1015 Mar 2012 [Refereed]

In the previous paper, the author gave linear inequalities on the coefficients of the Alexander polynomials of alternating knots of genus two, which are best possible as linear inequalities on the coefficients of them. In this paper, we give infin...

Proceedings of the japan Academy, Series A, Mathematical Sciences 87(2) 17-21 Feb 2011 [Refereed]

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 A...

Journal of Knot Theory and its Ramifications 19(8) 1075-1092 Aug 2010 [Refereed]

We give a family of linear inequalities which strictly estimate relation among the coefficients of the Alexander polynomials of alternating knots of genus two. We also give such families for positive knots of genus two, and for homogeneous knots o...

Communications in Analysis and Geometry 18(3) 579-600 Jul 2010 [Refereed]

We show that if a Montesinos knot admits a Dehn surgery yielding a toroidal Seifert fibered 3-manifold, then the knot is the trefoil knot and the surgery slope is 0.

Osaka Journal of Mathematics 46(2) 353-371 Jun 2009 [Refereed]

We confirm R.H. Fox's trapezoidal conjecture for alternating knots of genus two by a method different from P. Ozsváth and Z. Szabó's one. As an application, we determine the alternating knots of genus two whose Alexander polynomials have minimal c...

The cosmetic surgery conjecture saids that no pair of Dehn surgeries along inequivalent slopes yield
orientation preservingly homeomorphic 3-manifolds. First I will tlak about a recent result on this conjecture
for certain two-bridge knots. Next I...

The 11th East Asian School of Knots and Related Topics 28 Jan 2016

A pair of Dehn fillings on a 3-manifold with a torus boundary
are called chirally cosmetic if the obtained pair of manifolds are orientation
reversingly homeomorphic. Cosmetic fillings on a 3-manifold are said to be
exotic if there is no self-home...