Exceptional surgeries on alternating knots

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Exceptional surgeries on alternating

knots H.Masai

Introduction Exceptional surgery t(d)>9 arborescent knots Remaining cases

Using Computer Computation time

Exceptional surgeries on alternating knots

Hidetoshi Masai

Tokyo Institute of Technology

joint work with

Kazuhiro Ichihara (Nihon University)

Low-dimensional Geometry and Topology Tokyo Inst of Tech, Sep 13, 2012

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Motivation

Hyperbolic Dehn Surgery Theorem [Thurston (1978)]

The number of Dehn surgeries on ahyperbolicknot (i.e., knot with hyperbolic complement) yielding a non-hyperbolic manifold isFINITE.

Exceptional surgery

A Dehn surgery on a hyperbolicknot yielding a non-hyperbolicmanifold is calledexceptional.

Goal

Classify all the exceptional surgeries

on hyperbolic alternating knots.

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knots H.Masai

Motivation

Hyperbolic Dehn Surgery Theorem [Thurston (1978)]

The number of Dehn surgeries on ahyperbolicknot (i.e., knot with hyperbolic complement) yielding a non-hyperbolic manifold isFINITE.

Exceptional surgery

A Dehn surgery on a hyperbolicknot yielding a non-hyperbolicmanifold is calledexceptional.

Goal

Classify all the exceptional surgeries on hyperbolic alternating knots.

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Upper Bound of Twist Number

Fact [Lackenby (2000)]

If a hyperbolic alternating knotK has a connected prime alternating diagramD satisfying t(D)≥9,then K admits noexceptional surgeries.

Let Dbe a connected alternating diagram of a knot in S³ The twist numbert(D) is the number oftwists,

i.e., maximal connected collections of bigon regions inD or isolated crossings adjacent to no bigon regions.

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alternating arborescent knot

“Most” alternating knots with t(D)≤8 are arborescent.

An arborescent knotis a knot obtained by summing and gluing several rational tangles together.

Fact

All the exceptional surgeries on hyperbolic alternating arborescentknots other than P(2, p, q)or P(3,4,5)are completely classified

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Remaining cases

Lemma

If a hyperbolic non-arborescent alternating knotK has a connected prime alternating diagram Dsatisfying t(D)≤8, then K is obtained by twisting along some augmented loops from the knot 818 or the Borromean rings6³₂, or the link with at most 2 twist regions add to 6³₂.

Such knots can be obtained by (±1, n) surgeries onthe following links...

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Introduction Exceptional surgery t(d)>9 arborescent knots Remaining cases Using Computer

Computation time

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Computation time

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Computation time

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Computation time

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P(3,4,5), P(2, p, q)

We used a computer program;

B.Martelli, C.Petronio, F.Roukema Exceptional Dehn surgery on

the minimally twisted five-chain link preprint, arXiv:1109.0903

Available athttp://www.dm.unipi.it/~martelli/Dehn.html

Result of Computation

I The exceptional surgeries on alternating pretzel knots P(3,4,5), P(2, p, q) are completely classified.

I Every alternating non-arborescent hyperbolic knot with a diagram Dof t(D)≤9 hasnoexceptional surgeries.

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P(3,4,5), P(2, p, q)

We used a computer program;

B.Martelli, C.Petronio, F.Roukema Exceptional Dehn surgery on

the minimally twisted five-chain link preprint, arXiv:1109.0903

Available athttp://www.dm.unipi.it/~martelli/Dehn.html

Result of Computation

I The exceptional surgeries on alternating pretzel knots P(3,4,5), P(2, p, q) are completely classified.

I Every alternating non-arborescent hyperbolic knot with a diagram Dof t(D)≤9 hasnoexceptional surgeries.

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Procedure

The program relies upon

I SnapPy(based on SnapPea): computer software calculates various hyperbolic invariants for 3-manifolds.

To get a rigorous proof we need to apply Moser’s algorithm.

Harriet H. Moser

Proving a manifold to be hyperbolic once it has been approximated to be so Algebraic & Geometric Topology 9 (2009) 103–133.

It can give us a complete classificationof exceptional surgeries on a given hyperbolic link if we are lucky.

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Procedure

The program relies upon

I SnapPy(based on SnapPea): computer software calculates various hyperbolic invariants for 3-manifolds.

To get a rigorous proof we need to apply Moser’s algorithm.

Harriet H. Moser

Proving a manifold to be hyperbolic once it has been approximated to be so Algebraic & Geometric Topology 9 (2009) 103–133.

It can give us a complete classificationof exceptional surgeries on a given hyperbolic link if we are lucky.

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Computation time

I We have more than25,000 links to investigate.

I For ONE link, we need around2 HOURs.

⇒ We need to reduce the computation time.

Fact [Wu] + Observation Suppose

I the edges go through a crossing circle are anti-parallel

I After ”smoothing”,](knot components)>1.

⇒ Links obtained by Dehn surgeries on the crossing circle along slopes other than (±1,1)have NOexceptional surgeries.

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TSUBAME

I TSUBAME is the supercomputerof Tokyo Tech.

I Intuitively we can use manymachines simultaneously.

I I ”rent” 640 machines.

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So far we have applied code by Martelli-Petronio-Roukema for all our links and found NOexceptional fillings.

⇒

After we

I check the hyperbolicity of all triangulations by Moser’s algorithm.

we get a rigorous (up to floating point error) proof.

Floating point error

I Not every decimal number can be represented by a finite binary.

I For computation we need to give up certain error. This error is called floating point errror

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So far we have applied code by Martelli-Petronio-Roukema for all our links and found NOexceptional fillings.

⇒ After we

I check the hyperbolicity of all triangulations by Moser’s algorithm.

we get a rigorous (up to floating point error) proof.

Floating point error

I Not every decimal number can be represented by a finite binary.

I For computation we need to give up certain error.

This error is calledfloating point errror

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Exceptional surgeries on alternating knots

Exceptional surgeries on alternating knots

Classify all the exceptional surgeries

on hyperbolic alternating knots.

Classify all the exceptional surgeries on hyperbolic alternating knots.

Thank you for your attention!