Katura Miyazaki

Seifert-fibered surgeries which do not arise from primitive/Seifert-fibered constructions

We construct two infinite families of knots each of which admits a Seifert fibered surgery with none of these surgeries coming from Deans primitive/Seifert-fibered construction. This disproves a conjecture that all Seifert-fibered surgeries arise from Deans primitive/Seifert-fibered construction. The (-3, 3,5)-pretzel knot belongs to both of the infinite families.

Toroidal and annular Dehn surgeries of solid tori

Abstract We obtain an infinite family of hyperbolic knots in a solid torus which admit half-integral, toroidal and annular surgeries. Among this family we find a knot with two toroidal and annular surgeries; one is integral and the other is half-integral, and their distance is 5. This example realizes the maximal distance between annular surgery slopes and toroidal ones, and that between annular surgery slopes.

Band-sums are ribbon concordant to the connected sum

We show that an arbitrary band-connected sum of two or more knots are ribbon concordant to the connected sum of these knots. As an application we consider which knot can be a nontrivial band-connected sum.

Some well-disguised ribbon knots

Abstract For certain knots J in S 1 × D 2 , the dual knot J ∗ in S 1 × D 2 is defined. Let J ( O ) be the satellite knot of the unknot O with pattern J , and K be the satellite of J ( O ) with pattern J ∗ . The knot K then bounds a smooth disk in a 4-ball, but is not obviously a ribbon knot. We show that K is, in fact, ribbon. We also show that the connected sum J(O) # J ∗ (O) is a nonribbon knot for which all known algebraic obstructions to sliceness vanish.

Seifert fibering surgery on periodic knots

Abstract We show that if a periodic knot K in the 3-sphere yields a Seifert fibered manifold by Dehn surgery, then the quotient of K by the group action generated by any periodic map of K is a torus knot, except for a special case. We also consider what Seifert fibered manifolds are obtained by Dehn surgery on periodic knots. If a non-torus, periodic knot yields a Seifert fibered manifold M , then the base space of M is the 2-sphere; and some pair of exceptional fibers in M has indices coprime provided that M contains at most three exceptional fibers.

Conjugation and the prime decomposition of knots in closed, oriented 3-manifolds

In this paper we consider the prime decomposition of knots in closed, oriented 3-manifolds. (For classical knots one can easily prove the uniqueness of prime decomposition by using a standard innermost disk argument.) We define a new relation, conjugation, between oriented knots in closed, oriented 3-manifolds and prove the following results. ( 1 ) The prime decomposition is, roughly speaking, uniquely determined up to conjugation, (2) there is a prime knot á? in SlxS2 suchthat âl#3?\ = 3¡#X2 if ^¡ is a conjugation of ^2 ) and (3) if a knot 3? has a prime decomposition which does not contain 31, then it is the unique prime decomposition of X .

Networking Seifert surgeries on knots, III

How do Seifert surgeries on hyperbolic knots arise from those on torus knots? We approach this question from a networking viewpoint introduced in [9]. The Seifert surgery network is a 1‐dimensional complex whose vertices correspond to Seifert surgeries; two vertices are connected by an edge if one Seifert surgery is obtained from the other by a single twist along a trivial knot called a seiferter or along an annulus cobounded by seiferters. Successive twists along a “hyperbolic seiferter” or a “hyperbolic annular pair” produce infinitely many Seifert surgeries on hyperbolic knots. In this paper, we investigate Seifert surgeries on torus knots that have hyperbolic seiferters or hyperbolic annular pairs, and obtain results suggesting that such surgeries are restricted. 57M25; 57M50, 57N10 Dedicated to Sadayoshi Kojima on the occasion of his 60 th birthday

KNOTS THAT HAVE MORE THAN TWO ACCIDENTAL SLOPES

An accidental surface in the exterior of a knot in the 3-sphere is a closed essential surface for which there is an annulus in the knot exterior X connecting a loop in the surface and a nontrivial loop in ∂X, the peripheral torus of the knot. The isotopy class of the loop in ∂X is called an accidental slope; each accidental surface has a unique accidental slope. It is known that accidental slopes are integral or 1/0, and there is a knot with two accidental slopes 0 and 1/0. We show that for any integer m≥0, there is a hyperbolic knot which has m+1 accidental surfaces with accidental slopes 0,1,…,m.

Crossing change and exceptional Dehn surgery

Thurston’s hyperbolic Dehn surgery theorem [11], [12] asserts that if a knot in the 3-sphere 3 is hyperbolic (i.e., 3 − admits a complete hyperbolic structure of finite volume), then all but finitely many Dehn surgeries on yield hyperbolic 3-manifolds. By an exceptional surgery on a hyperbolic knot we mean a nontrivial Dehn surgery producing a non-hyperbolic manifold. Refer to [3], [6] for a survey on Dehn surgery on knots. We empirically know that ‘most’ knots are hyperbolic and ‘most’ hyperbolic knots have no exceptional surgeries. In this paper, we demonstrate the abundance of hyperbolic knots with no exceptional surgeries by showing that every knot is ‘close’ to infinitely many such hyperbolic knots in terms of crossing change. We regard that two knots are the same if they are isotopic in 3. For a knot in 3, let ( ) be the set of knots each of which is obtained by changing at most crossings in a diagram of .