Combinatorial suspension for disc-homeomorphisms

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HAL Id: hal-00914471

https://hal.archives-ouvertes.fr/hal-00914471

Submitted on 5 Dec 2013

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Combinatorial suspension for disc-homeomorphisms

François Gautero, Jérôme Los

To cite this version:

François Gautero, Jérôme Los. Combinatorial suspension for disc-homeomorphisms. Journal of Knot Theory and Its Ramifications, World Scientific Publishing, 1998, 7 (6), pp.747-795. �hal-00914471�

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A vertex vAn edge e

The polygon P(v)

The rectangle R(e) A tie e(e) A P -side_v

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A path in N(K)

A tie e(e)

A collection of paths

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The initial axis

The terminal axis

1 2 3 1 2 3 x y z 1 2 3 1 2 3

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Type

I

II Type

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1 2 3 4 5 Q Q ’ P P ’ F₁ S ’₁ S1 F₅

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P_i P_i+1

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1 2 3 4 3 4 1 2

conjugacy

X X

X not accessible

X accessible

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P C1 _C 2 i(P(v)E P) C E ,P(v)1 C E ,P(v)2

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A G-point

Toward

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A tie e(e) A rectangle R(e) orientation of the edge e Interval in ( K ) E R(e) a nailed point

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R(e) R(d) (e) = d (e) = d -1 R(d) Paths in F(R ) Intervals on (K) the accordion curve

K

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I I I I (e) i i+1 i+2 Bj Bm Bt Bj Bm _B t J(e) Ji Ji+1 Ji+2 Bi Bi Bi+1 Bi+1 Bi+2 Bi+2 D x {1} D x {0} 2 2 a) b) c) + d) Bj Bm Bt Bi Bi+1 _B i+2 J I (e) (e)

locally k-sheeted side

locally one sheeted side (k = 3)

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Accordion curve R Projection of R to the initial axis

1 2 3 4 I _{II III IV} 2 1 3 4 initial axis I II III IV

The braid carried by the branched surface

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Paths in BP

Points in P The accordion curve

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( )

a

b

c

The braid `

a

b

c

1

2

3

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Terminal side a G-point _{a generating} path a boundary path (a)

the accordion curve the images of the boundary paths intersection points

nailed points

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1 2 3 4 5 6 7 8 9

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A

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