Unknotting operations for bered knots and pseudo- ber surfaces€¦ · Yukari Funakoshi Unknotting...

Transcript of Unknotting operations for bered knots and pseudo- ber surfaces€¦ · Yukari Funakoshi Unknotting...

. . . . . .

Introduction and PreliminariesPre-fiber surface

pseudo-fiber surface of level n

.

......

Unknotting operations for fibered knots andpseudo-fiber surfaces

Yukari Funakoshi

Nara Women’s University

December 21, 2013

Yukari Funakoshi Unknotting operations for fibered knots and pseudo-fiber surfaces
. . . . . .



.Definition..

......

n-component link : S11 ∪ S12 ∪ · · · ∪ S1n ↪→ S3mutually disjoint oriented 1-spheres embedded in S3

knot: one component link

.Example..

......

. . . . . .



.Definition (unknotting operation)..

......

.Definition..

......

K :knot,u(K ):=min{ the number of unknotting operations

required to obtain a trivial knot from K }: unknotting number of K

. . . . . .



.. Sharlemann-Thompson’s work on unknotting number

L: link.Definition (Seifert surface)..

......

A Seifert surface S for L isa compact oriented surface w/o closed comp. (⊂ S3) s.t. ∂S = L

Fact. There exists Seifert surface, for ∀L..Definition (minimal genus Seifert surface, genus of a knot)..

......

The genus of a knot K isg(K ) := min{genus(F )|F : Seifert surface for K}A Seifert surface S for K isminimal genus if genus(S) = g(K ).

Note. K : trivial knot ⇔ g(K ) = 0

. . . . . .



.Theorem (Scharlemann-Thompson)..

......

L, L′: knotL → L′: unknotting operationg(L′) < g(L)∃S : minimal genus Seifert surface of L

s.t. S is a plumbing of a surface and Hopf band,

and

M. Sharlemann, A. Thomnpson, Link genus and Conway moves,

Comment. Math. Helv. 64(1989), no.4, 527-53Yukari Funakoshi Unknotting operations for fibered knots and pseudo-fiber surfaces
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.Definition (sutured manifold)..

......

(M, γ)is a sutured manifold.

⇐⇒defM: an oriented 3−manifold

γ : union of annuli (⊂ ∂M)

with s(γ) : oriented core curves of γ ,where each component of R(γ) := ∂M \ Intγ is orientedso that the orientation meets s(γ).

(Each component of s(γ) is called a suture.)

. . . . . .



Fix an orientation of M. Then:

R+(γ)(R−(γ) resp.)

:=the union of the component of R(γ)whose normal vector points out (into resp.) M.

. . . . . .



.Example (product sutured manifold)..

......

S : compactorientable surface

M = S × [0, 1]γ = ∂S × [0, 1]

Then (M, γ) is a sutured manifolds.t. R+(γ) = S × {1}

R−(γ) = S × {0}

Such (M, γ) is called a product sutured manifold.

. . . . . .



L : linkS : Seifert surface for LE (L)(:= S3 \ IntN(L))

: exterior of L(For simplicitywe denote S ∩ E (L) by S .)N = N(S ;E (L))δ = N ∩ ∂E (L)

The product sutured manifold (N, δ) is calledthe sutured manifold obtained from S .

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.Definition (complementary sutured manifold)..

......

Nc = cl(E (L) \ N)δc = cl(∂E (L) \ δ)R±(δ

c) = R∓(δ)

(Nc , δc) is called the complementary sutured manifold for S.

. . . . . .



L : link, S : Seifert surface for L.Definition (fiber surface)..

......

S is a fiber surface , if :the complementary sutured manifold

is a product sutured manifold.

.Definition (fibered link)..

......

L is a fibered link if∃S : Seifert surface for L s.t. S is a fiber surface.

.Example..

......

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T. Kobayashi, Fiberd links and unknotting operations,Osaka J. Math. 26(1989), 699-742

L: linkS : Seifert surface for L(Nc , δc):

complementary sutured manifold for S.Definition (pre-fiber surface)..

......

S is a pre-fiber surface, if:∃D±(⊂ Nc): mutually disjoint disks

s.t. R±(δc)∩D± = ∂D±: essential in R±(δc)

(such disk is called a compressing disk for R±(δc)),

and (Nc ′, δc) is a product sutured manifold,

where Nc ′ is a 3-manifold obtained from Nc

by cutting along D+ ∪ D−.

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. . . . . .



.. Examples of pre-fiber surfaces

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.Theorem (Kobayashi)..

......

Suppose L: fibered knot s.t.u(L) = 1.

L: fibered L′: trivial

S : fiber surface

Then S ′ is a pre-fiber surface.

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.Theorem..

......

S :pre-fiber surfacewith canonical compressing disks D̄+,D̄−.

Let α be an arc properly embedded in S .s.t. α ∩ D̄+: 1-point,

α ∩ D̄−: 1-point.

The surface obtained from Sby applying a twist along α is a fiber surface.

. . . . . .



.. Motivation:

Generalize the above results

for fibered knots with unknotting numbers > 1.

. . . . . .



.. A generalization of pre-fiber surface.

Let L: linkS : Seifert surface

(Nc , δc): complementary sutured manifold for S

.Definition (pseudo-fiber surface of level n)..

......

・・・

・・・

For n ≥ 0, S is a pseudo-fiber surface of level n, if:

∃D±1 , . . . ,D±n:mutually disjoint compressing disks of R±(δ

c)

s.t. (Nc ′, δc)is a product sutured manifold,where Nc ′ is a 3-manifold obtained from Nc

by cutting alongD+1 ∪ · · · ∪ D+n ∪ D

−1 ∪ · · · ∪ D−n

S is a pseudo-fiber surface of level 0 ⇔ S is a fiber surfaceS is a pseudo-fiber surface of level 1 ⇔ S is a pre-fiber surface

. . . . . .



.. A generalization of Theorem (Kobayashi)


......

S :pre-fiber surface with canonical compressing disks D̄+,D̄−.Let α be an arc properly embedded in S s.t. α ∩ D̄+: 1-point,

α ∩ D̄−: 1-point.The surface obtained from S by applying a twist along α is a fiber surface.

.Theorem..

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Let S : a pseudo-fiber surface of level nα1, . . . , αp: mutually disjoint arcs properly embedded in S

(p ≤ n)Suppose that ∃D̄+1 , . . . , D̄+n , D̄−1 , . . . , D̄

−n

: a system of canonical compressing diskss.t. αi ∩ D̄±j = ∅ for ∀i , j (i ̸= j),

αi ∩ ∂D̄+i (∂D̄−i resp.): 1-point. (i = 1, . . . , p)

Thenthe surface obtained from S by twisting along α1 ∪ · · · ∪ αpis a pseudo-fiber surface of level n − p.

. . . . . .




......

Suppose L: fibered knot s.t.u(L) = 1.

S : fiber surface S ′: pre-fiber surface

.Definition (Ascending sequence of pseudo-fiber surfaces)..

......

A sequence of pseudo-fiber surfaces Si of level piS0 → S1 → · · · → Si → Si+1 → · · · → Sn

where pi ≤ pi+1 (i = 0, 1, . . . , n − 1)p0 = 0

is an ascending sequence of pseudo-fiber surface if:

∃α(i)1 , . . . , α(i)qi : mutually disjoint arcs properly embedded in Si

s.t. Si+1 is obtained from Si by twisting along α(i)1 ∪ · · · ∪α

(i)qi .

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.Question..

......

For each fibered knot K ,

does there exist an ascending sequence of pseudo-fiber surfacesrealizing the unknotting number u(K )?

More precisely,does there exist an ascending sequence of pseudo-fiber surfaces

S0 → S1 → · · · → Sn

(with levels pi , arcs α(i)1 , . . . , α

(i)qi as above)

such that ∂S0 = K , ∂Sn = trivial knot, andu(K ) = q0 + q1 + · · ·+ qn−1?

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.. Torus knot

Notation:T : unknotted torus(⊂ S3)

Fix an oriented meridian curve, andan oriented longitude curve on T .

Let p, q: relatively prime integers,p, q ̸= 0

.Definition (Torus knot of type (p, q))..

......

T (p, q) is a knot embedded in T which wrapsp times along the meridian andq times along the longitude.Such T (p, q) is called a torus knot of type (p, q).

It is known that each torus knot is a fibered knot.

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.Theorem 2.1..

......

...1 A torus knot T (p, q) is trivial if and only ifeither p = ±1 or q = ±1.

...2 Two non-trivial torus knotsT (p, q) and T (p′, q′) are equivalent if and only if(p′, q′) is equal to one of the following,

(±p,±q), (±q,±p), (±p,∓q), (±q,∓p).

T (p, q): torus knot of type (p, q), where 0 < p < q.Theorem (Kronheimer-Mrowka)..

......u(T (p, q)) =

(p − 1)(q − 1)2

Kronheimer, P. B.; Mrowka, T. S. , Gauge theory for embedded

surfaces, I, Topology 32 (1993), no.4, 773―826Yukari Funakoshi Unknotting operations for fibered knots and pseudo-fiber surfaces
. . . . . .



.. Consider the fractional expansion (0 < p < q)

q

p= a0 +

1

a1 +1

. . .1

am−1 +1

pm−1

, (ai > 0, pm−1 > 1)

Euclidian algorithmpi−1 = piai+1 + pi+1(0 ≤ i ≤ m − 2, p−1 = q, p0 = p, 0 < pi+1 < pi )

7

5= 1 +

2

5= 1 +

1

5

2

= 1 +1

2 +1

2

(7 = 5× 1 + 2) (5 = 2× 2 + 1)Yukari Funakoshi Unknotting operations for fibered knots and pseudo-fiber surfaces
. . . . . .



.Theorem 5.1..

......

T (p, q): torus knot of type (p, q)∃ S0 → S1 → S2 → · · · → Sm

: an ascending sequence of pseudo-fiber surfaces for T (p, q)realizing the unknotting number of T (p, q) with length m,

s.t. the level of Si is∑i

k=0(pk−1 − 1)ak−1.

.Example 5.2..

......

Note that u(T (5, 7)) = 12, Note7

5= 1 +

1

2 +1

2∃ S0 → S1 → S2 : ascending sequence

realizing the unknotting number of T (5, 7),where the level of S1 is 4,

the level of S2 is 6.

. . . . . .



Notation.

Bn: n-braid groupσ1, . . . , σn−1: standard generaters of Bnβ ∈ Bn ⇒ β̂: closure of β

Fact.

T (p, q) = ̂(σ1 · · ·σq−1)p

. . . . . .



Recall:q

p= a0 +

p1

p(a0 > 0, 0 < p1 < p, 0 < p < q)

.Key Fact 1..

......

a0 ×p(p − 1)

2simultaneous unknotting operations

T (p, q) −→−→−→−→−→−→−→−→−→−→−→ T (p1, p)

This Fact is proved later.

. . . . . .



Recall:q

p= a0 +

p1

p(a0 > 0, 0 < p1 < p, 0 < p < q)

.Key Fact 1..

......

a0 ×p(p − 1)

2simultaneous unknotting operations

T (p, q) −→−→−→−→−→−→−→−→−→−→−→ T (p1, p)

This Fact is proved later.

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Note that: by Kronheimer-Mrowka,u(T (p, q))− u(T (p1, p))

=(p − 1)(q − 1)

2− (p1 − 1)(p − 1)

2= a0 ×

p(p − 1)2

By this observation together with Fact, we seeu(T (p, q)) is realized byT (p, q) → T (p, p1) = T (p1, p) → T (p1, p2) = T (p2, p1) → · · ·

a0p(p − 1)

2a1

p1(p1 − 1)2

· · · → T (pm−1, 1): trivial knotWe will see this sequence gives the ascending sequence ofpseudo-fiber surface in Theorem 5.1.

. . . . . .



Note that: by Kronheimer-Mrowka,u(T (p, q))− u(T (p1, p))

=(p − 1)(q − 1)

2− (p1 − 1)(p − 1)

2= a0 ×

p(p − 1)2

By this observation together with Fact, we seeu(T (p, q)) is realized byT (p, q) → T (p, p1) = T (p1, p) → T (p1, p2) = T (p2, p1) → · · ·

a0p(p − 1)

2a1

p1(p1 − 1)2

· · · → T (pm−1, 1): trivial knot

.Key Fact 2........The twists in Key Fact 1 produces a pseudo-fiber surface.

. . . . . .



Let β = σϵ1i1 σϵ2i2· · ·σϵmim (∈ Bn), ϵi = ±1

Sβ :=Seifert surface obtained from β̂by applying Seifert algorithm.

⇔ Sβ is obtained from n horizontal disks by attaching m half twistbands according to the representative σϵ1i1 σ

ϵ2i2· · ·σϵmim .

Notation.

τ ϵ1i1 τϵ2i2· · · τ ϵmim denotes Sβ,

where τϵjij

denotes the half twist band corresponding to σϵjij

Let γ = σν1j1 σν2j2· · ·σνℓjℓ ∈ Bq be another braid such that γ̄ = β̄. We

say: Sβ = τϵ1i1τ ϵ2i2 · · · τ

ϵmim

and Sγ = τν1j1τν2j2 · · · τ

νℓjℓ

are equivalent

(denoted by τ ϵ1i1 τϵ2i2· · · τ ϵmim = τ

ν1j1τν2j2 · · · τ

νℓjℓ).

if there is an ambient isotopy of S3 transforming Sβ to Sγ .

. . . . . .



.. 関係式の例

A,B: words in τ ’s.lemma4.1..

......

Sβ = A · τ ϵii τϵjj · B: be the Seifert surface for β̄.

Suppose |i − j | > 1.Then A · τ ϵii τ

ϵjj · B = A · τ

ϵjj τ

ϵii · B.

.Lemma..

......

Sβ = A · τ±1i τi+1τi · B: be the Seifert surface for β̄.Then we have A · τ±1i τi+1τi · B = A · τi+1τiτ

±1i+1 · B

. . . . . .



.. Extended Notation.

τ−1i τi in A · τ−1i τi · B

does not give a “trivial part”,but a tube attached to A · B.

We shall denote the subsurface as in Figure by

(i + 1i

),

and the Seifert surface in the right side of Figure by A

(i + 1i

)B.

A · τ−1i+1τ−1i τiτi+1 · B A

(i + 1 i + 2i i + 1

)B

. . . . . .



.. 関係式の例

.lemma4.3..

......

For each j ∈ {1, . . . , n − 1} \ {k, k + 1}, we have

A ·(

i + 1 · · · k + 1i · · · k

)τ±1j ·B=A ·τ

±1j

(i + 1 · · · k + 1i · · · k

)·B

. . . . . .



.. 関係式の例.lemma4.4..

......

For each pair r , k (2 ≤ r ≤ k ≤ p − 1), we have

Aτr−1 · · · τk(

r · · · kr − 1 · · · k − 1

)B

= A

(r + 1 · · · k + 1r · · · k

)τr−1 · · · τkB.

. . . . . .




......

For any pairs of pair of integer(r , s) with (2 ≤ r ≤ s ≤ p) and (t, u) with (2 ≤ t ≤ u ≤ p),

we have A

(t · · · u

t − 1 · · · u − 1

)(r · · · s

r − 1 · · · s − 1

)B

= A

(r · · · s

r − 1 · · · s − 1

)(t · · · u

t − 1 · · · u − 1

)B

＜ s ̸= u＞

...

...Yukari Funakoshi Unknotting operations for fibered knots and pseudo-fiber surfaces
. . . . . .




......

For any pairs of pair of integer(r , s) with (2 ≤ r ≤ s ≤ p) and (t, u) with (2 ≤ t ≤ u ≤ p),

we have A

(t · · · u

t − 1 · · · u − 1

)(r · · · s

r − 1 · · · s − 1

)B

= A

(r · · · s

r − 1 · · · s − 1

)(t · · · u

t − 1 · · · u − 1

)B

＜ s = u＞

...

......

...

...

...

...

...

...

...

...

...

. . . . . .



. . . . . .



(σ1σ2σ3σ4)7

. . . . . .



(τ1τ2τ3τ4)7

. . . . . .



(τ1τ2τ3τ4)7

. . . . . .



(τ1τ2τ3τ4)7

. . . . . .



(τ1τ2τ3τ4)(τ1τ2τ3τ4)(τ1τ2τ3τ4)(τ1τ2τ3τ4)(τ1τ2τ3τ4)3

. . . . . .



(τ−11 τ−12 τ

−13 τ

−14 )(τ

−11 τ

−12 τ

−13 τ4)(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

. . . . . .



(τ−11 )(τ−12 τ

−13 τ

−14 )(τ

−11 τ

−12 τ

−13 τ4 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

. . . . . .



(τ−11 τ1 )(τ−12 τ

−13 τ

−14 )(τ

−11 τ

−12 τ

−13 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

. . . . . .



(τ−11 τ−12 τ

−13 τ

−14 )(τ

−11 τ

−12

::::::τ−13 τ4)(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 τ

−12 τ

−13 )(τ

−11 τ

−12

::::::)(τ−14 τ

−13 τ4 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 τ

−12 τ

−13 )(τ

−11 τ

−12 )( τ3 τ

−14 τ

−13 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 τ

−12 τ

−11 )(τ

−13 τ

−12 τ3 )(τ

−14 τ

−13 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 τ

−12 τ

−11 )( τ2 τ

−13 τ

−12 )(τ

−14 τ

−13 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 )(τ

−12 τ

−11 τ2 )(τ

−13 τ

−12 )(τ

−14 τ

−13 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 )( τ1 τ

−12 τ

−11 )(τ

−13 τ

−12 )(τ

−14 τ

−13 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

. . . . . .



(τ−11 τ−12 τ

−13 τ

−14 )(τ

−11 τ

−12

::::::τ−13 τ4)(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 τ

−12 τ

−13 )(τ

−11 τ

−12

::::::)(τ−14 τ

−13 τ4 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 τ

−12 τ

−13 )(τ

−11 τ

−12 )( τ3 τ

−14 τ

−13 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 τ

−12 τ

−11 )(τ

−13 τ

−12 τ3 )(τ

−14 τ

−13 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 τ

−12 τ

−11 )( τ2 τ

−13 τ

−12 )(τ

−14 τ

−13 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 )(τ

−12 τ

−11 τ2 )(τ

−13 τ

−12 )(τ

−14 τ

−13 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 )( τ1 τ

−12 τ

−11 )(τ

−13 τ

−12 )(τ

−14 τ

−13 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

. . . . . .



(τ−11 τ−12 τ

−13 τ

−14 )(τ

−11 τ

−12

::::::τ−13 τ4)(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 τ

−12 τ

−13 )(τ

−11 τ

−12

::::::)(τ−14 τ

−13 τ4 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 τ

−12 τ

−13 )(τ

−11 τ

−12 )( τ3 τ

−14 τ

−13 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 τ

−12 τ

−11 )(τ

−13 τ

−12 τ3 )(τ

−14 τ

−13 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 τ

−12 τ

−11 )( τ2 τ

−13 τ

−12 )(τ

−14 τ

−13 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 )(τ

−12 τ

−11 τ2 )(τ

−13 τ

−12 )(τ

−14 τ

−13 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 )( τ1 τ

−12 τ

−11 )(τ

−13 τ

−12 )(τ

−14 τ

−13 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

. . . . . .



(τ−11 τ−12 τ

−13 τ

−14 )(τ

−11 τ

−12

::::::τ−13 τ4)(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 τ

−12 τ

−13 )(τ

−11 τ

−12

::::::)(τ−14 τ

−13 τ4 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 τ

−12 τ

−13 )(τ

−11 τ

−12 )( τ3 τ

−14 τ

−13 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 τ

−12 τ

−11 )(τ

−13 τ

−12 τ3 )(τ

−14 τ

−13 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 τ

−12 τ

−11 )( τ2 τ

−13 τ

−12 )(τ

−14 τ

−13 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 )(τ

−12 τ

−11 τ2 )(τ

−13 τ

−12 )(τ

−14 τ

−13 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 )( τ1 τ

−12 τ

−11 )(τ

−13 τ

−12 )(τ

−14 τ

−13 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

. . . . . .



(τ−11 τ−12 τ

−13 τ

−14 )(τ

−11 τ

−12

::::::τ−13 τ4)(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 τ

−12 τ

−13 )(τ

−11 τ

−12

::::::)(τ−14 τ

−13 τ4 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 τ

−12 τ

−13 )(τ

−11 τ

−12 )( τ3 τ

−14 τ

−13 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 τ

−12 τ

−11 )(τ

−13 τ

−12 τ3 )(τ

−14 τ

−13 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 τ

−12 τ

−11 )( τ2 τ

−13 τ

−12 )(τ

−14 τ

−13 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 )(τ

−12 τ

−11 τ2 )(τ

−13 τ

−12 )(τ

−14 τ

−13 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 )( τ1 τ

−12 τ

−11 )(τ

−13 τ

−12 )(τ

−14 τ

−13 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

. . . . . .



(τ−11 τ−12 τ

−13 τ

−14 )(τ

−11 τ

−12

::::::τ−13 τ4)(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 τ

−12 τ

−13 )(τ

−11 τ

−12

::::::)(τ−14 τ

−13 τ4 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 τ

−12 τ

−13 )(τ

−11 τ

−12 )( τ3 τ

−14 τ

−13 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 τ

−12 τ

−11 )(τ

−13 τ

−12 τ3 )(τ

−14 τ

−13 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 τ

−12 τ

−11 )( τ2 τ

−13 τ

−12 )(τ

−14 τ

−13 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 )(τ

−12 τ

−11 τ2 )(τ

−13 τ

−12 )(τ

−14 τ

−13 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 )( τ1 τ

−12 τ

−11 )(τ

−13 τ

−12 )(τ

−14 τ

−13 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

. . . . . .



(τ−11 τ−12 τ

−13 τ

−14 )(τ

−11 τ

−12

::::::τ−13 τ4)(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 τ

−12 τ

−13 )(τ

−11 τ

−12

::::::)(τ−14 τ

−13 τ4 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 τ

−12 τ

−13 )(τ

−11 τ

−12 )( τ3 τ

−14 τ

−13 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 τ

−12 τ

−11 )(τ

−13 τ

−12 τ3 )(τ

−14 τ

−13 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 τ

−12 τ

−11 )( τ2 τ

−13 τ

−12 )(τ

−14 τ

−13 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 )(τ

−12 τ

−11 τ2 )(τ

−13 τ

−12 )(τ

−14 τ

−13 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

↓(τ−11 )( τ1 τ

−12 τ

−11 )(τ

−13 τ

−12 )(τ

−14 τ

−13 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

. . . . . .



(τ−11 )(τ−12 τ

−13 τ

−14 )(τ

−11 τ

−12 τ

−13 τ4 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

. . . . . .



(τ−11 τ1 )(τ−12 τ

−13 τ

−14 )(τ

−11 τ

−12 τ

−13 )(τ

−11 τ

−12 τ3τ4)(τ

−11 τ2τ3τ4)(τ1τ2τ3τ4)

3

. . . . . .



(τ−11 τ1 )(τ−12 τ

−13 τ

−14 )(τ

−11 )(τ

−12 τ

−13 )(τ

−11 τ

−12 τ3 τ4)(τ

−11 τ2τ3τ4)

(τ1τ2τ3τ4)3

. . . . . .



(τ−11 τ1 )(τ−12 τ

−13 τ

−14 )(τ

−11 τ1 )(τ

−12 τ

−13 )(τ

−11 τ

−12 τ4)(τ

−11 τ2τ3τ4)

(τ1τ2τ3τ4)3

. . . . . .



(τ−11 τ1 )(τ−12 τ

−13 τ

−14 )(τ

−11 τ1 )(τ

−12 τ

−13 )(τ

−11 )(τ

−12 τ4)(τ

−11 τ2 τ3τ4)

(τ1τ2τ3τ4)3

. . . . . .



(τ−11 τ1 )(τ−12 τ

−13 τ

−14 )(τ

−11 τ1 )(τ

−12 τ

−13 )(τ

−11 τ1 )(τ

−12 τ4)(τ

−11 τ3τ4)

(τ1τ2τ3τ4)3

. . . . . .



(τ−11 τ1 )(τ−12 τ

−13 τ

−14 )(τ

−11 τ1 )(τ

−12 τ

−13 )(τ

−11 τ1 )(τ

−12 τ4)

(τ−11 τ3τ4)(τ1τ2τ3τ4)(τ1τ2τ3τ4)2

. . . . . .



(τ−11 τ1 )(τ−12 τ

−13 τ

−14 )(τ

−11 τ1 )(τ

−12 τ

−13 )(τ

−11 τ1 )(τ

−12 τ4)

(τ−11 )(τ3τ4)( τ1 τ2τ3τ4)(τ1τ2τ3τ4)2

. . . . . .



(τ−11 τ1 )(τ−12 τ

−13 τ

−14 )(τ

−11 τ1 )(τ

−12 τ

−13 )(τ

−11 τ1 )(τ

−12 τ4)

(τ−11 τ1 )(τ3τ4)(τ2τ3τ4)(τ1τ2τ3τ4)2

. . . . . .



(21

)(τ−12 τ

−13 τ

−14 )

(21

)(τ−12 τ

−13 )

(21

)(τ−12 τ4)(

21

)(τ3τ4)(τ2τ3τ4)(τ1τ2τ3τ4)

2

. . . . . .



(21

)(τ−12 τ

−13 τ

−14 )

(21

)::::

(τ−12 τ−13 )

(21

)(τ−12 τ4 )(

21

)(τ3τ4)(τ2τ3τ4)(τ1τ2τ3τ4)

2

. . . . . .



(21

)(τ−12 τ

−13 τ

−14 )

(21

)::::

(τ−12 τ−13 τ4 )

(21

)(τ−12 )(

21

)(τ3τ4)(τ2τ3τ4)(τ1τ2τ3τ4)

2

. . . . . .



(21

)(τ−12

(21

)::::

)(τ−13 τ−14 )(τ

−12 τ

−13 τ4 )

(21

)(τ−12 )(

21

)(τ3τ4)(τ2τ3τ4)(τ1τ2τ3τ4)

2

. . . . . .



(21

)(τ−12

(21

)::::

τ2 )(τ−13 τ

−14 )(τ

−12 τ

−13 )

(21

)(τ−12 )(

21

)(τ3τ4)(τ2τ3τ4)(τ1τ2τ3τ4)

2

. . . . . .



(21

)·(2 31 2

)(τ−13 τ

−14 )(τ

−12 τ

−13 )

(21

)(τ−12 )(

21

)(τ3τ4)(τ2τ3τ4)(τ1τ2τ3τ4)

2

. . . . . .



(21

)·(2 31 2

)(τ−13 τ

−14 )(τ

−12 τ

−13 )

(21

)(τ−12 )

(21

)( τ3 τ4)(τ2τ3τ4)

(τ1τ2τ3τ4)2

. . . . . .



(21

)·(2 31 2

)(τ−13 τ

−14 )(τ

−12 τ

−13 )

(21

)::::

(τ−12 )

(21

)( τ3 τ4)(τ2τ3τ4)

(τ1τ2τ3τ4)2

. . . . . .



(21

)·(2 31 2

)(τ−13 τ

−14 )(τ

−12

(21

)::::

)(τ−13 )(τ−12 )

(21

)( τ3 τ4)(τ2τ3τ4)

(τ1τ2τ3τ4)2

. . . . . .



(21

)·(2 31 2

)(τ−13 τ

−14 )(τ

−12

(21

)::::

τ2 )(τ−13 )(τ

−12 )

(21

)(τ4)(τ2τ3τ4)

(τ1τ2τ3τ4)2

. . . . . .



(21

)·(2 31 2

)(τ−13 τ

−14 )(τ

−12

(21

)::::

τ2 )(τ−13 )(τ

−12 )

(21

)::::

(τ4)( τ2 τ3τ4)

(τ1τ2τ3τ4)2

. . . . . .



(21

)·(2 31 2

)(τ−13 τ

−14 )(τ

−12

(21

)::::

τ2 )(τ−13 )(τ

−12

(21

)::::

)(τ4)( τ2 τ3τ4)

(τ1τ2τ3τ4)2

. . . . . .



(21

)·(2 31 2

)(τ−13 τ

−14 )(τ

−12

(21

)::::

τ2 )(τ−13 )(τ

−12

(21

)::::

τ2 )(τ4)(τ3τ4)

(τ1τ2τ3τ4)2

. . . . . .



(21

)·(2 31 2

)(τ−13 τ

−14 )

(2 31 2

)(τ−13 )

(2 31 2

)(τ4)(τ3τ4)

(τ1τ2τ3τ4)2

. . . . . .



(21

)·(2 31 2

)(τ−13 τ

−14 )

(2 31 2

):::::::

(τ−13 )

(2 31 2

)( τ4 )(τ3τ4)

(τ1τ2τ3τ4)2

. . . . . .



(21

)·(2 31 2

)(τ−13

(2 31 2

):::::::

)(τ−14 )(τ−13 )( τ4 )

(2 31 2

)(τ3τ4)

(τ1τ2τ3τ4)2

. . . . . .



(21

)·(2 31 2

)(τ−13

(2 31 2

):::::::

τ3 )(τ−14 )(τ

−13 )

(2 31 2

)(τ3τ4)

(τ1τ2τ3τ4)2

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)(τ−14 )(τ

−13 )

(2 31 2

)(τ3τ4)

(τ1τ2τ3τ4)2

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)(τ−14 )(τ

−13

(2 31 2

)τ3)(τ4)

(τ1τ2τ3τ4)2

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)(τ−14 )

(2 3 41 2 3

)(τ4)

(τ1τ2τ3τ4)2

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)(τ−14

(2 3 41 2 3

)τ4)(τ1τ2τ3τ4)

2

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)·(2 3 4 51 2 3 4

):::::::::::::

· (τ1τ2τ3τ4)2

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)·(2 3 4 51 2 3 4

):::::::::::::

· (τ1τ2τ3τ4)2

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)·(2 3 4 51 2 3 4

):::::::::::::

· (τ1τ2τ3τ4)2

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)·(2 3 4 51 2 3 4

):::::::::::::

· (τ1τ2τ3τ4)2

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)·(2 3 4 51 2 3 4

):::::::::::::

· (τ1τ2τ3τ4)2

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)·(2 3 4 51 2 3 4

):::::::::::::

· (τ1τ2τ3τ4)2

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)·(2 3 4 51 2 3 4

):::::::::::::

· (τ1τ2τ3τ4)2

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)(τ1τ2τ3τ4)

2(2 3 4 51 2 3 4

):::::::::::::

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)(τ1τ2τ3τ4)

2(2 3 4 51 2 3 4

):::::::::::::

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)(τ1τ2τ3τ4)

2(2 3 4 51 2 3 4

):::::::::::::

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)(τ1τ2τ3τ4)

2(2 3 4 51 2 3 4

):::::::::::::

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)(τ1τ2τ3τ4)

2(2 3 4 51 2 3 4

):::::::::::::

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)(τ1τ2τ3τ4)

2(2 3 4 51 2 3 4

):::::::::::::

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)(τ1τ2τ3τ4)

2 (2 3 4 51 2 3 4

):::::::::::::

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)(τ1τ2τ3τ4)

2

(2 3 4 51 2 3 4

):::::::::::::

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)(τ1τ2τ3τ4)

2

(2 3 4 51 2 3 4

):::::::::::::

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)(τ1τ2τ3τ4)(τ1τ2τ3τ4)

(2 3 4 51 2 3 4

):::::::::::::

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)(τ1τ2τ3τ4)(τ1τ2τ3τ4)

(2 3 4 51 2 3 4

):::::::::::::

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)(τ1τ2τ3τ4)(τ1τ2τ3τ4)

(3 4 5 51 2 3 4

):::::::::::::

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)(τ1τ2τ3τ4)(τ1τ2τ3τ4)

(3 4 5 51 2 3 4

):::::::::::::

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)(τ1τ2τ3τ4)(τ1τ2τ3τ4)

(3 4 5 51 2 3 4

):::::::::::::

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)(τ1τ2τ3τ4)(τ1τ2τ3τ4)

(4 5 5 51 2 3 4

):::::::::::::

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)(τ1τ2τ3τ4)(τ1τ2τ3τ4)

(4 5 5 51 2 3 4

):::::::::::::

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)(τ1τ2τ3τ4)(τ1τ2τ3τ4)

(5 5 5 51 2 3 4

):::::::::::::

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)(τ1τ2τ3τ4)(τ1τ2τ3τ4)

(5 5 5 51 2 3 4

):::::::::::::

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)(τ1τ2τ3τ4)(τ1τ2τ3τ4)

(5 5 5 51 2 3 4

):::::::::::::

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)(τ1τ2τ3τ4)(τ1τ2τ3τ4)

(5 5 5 51 2 3 4

):::::::::::::

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)(τ1τ2τ3τ4)(τ1τ2τ3τ4)

(5 5 5 51 2 3 4

):::::::::::::

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)(τ1τ2τ3τ4)(τ1τ2τ3τ4)

(5 5 5 51 2 3 4

):::::::::::::

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)(τ1τ2τ3τ4)(τ1τ2τ3τ4)

(5 5 5 51 2 3 4

):::::::::::::

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)(τ1τ2τ3τ4)(τ1τ2τ3τ4)

(5 5 5 52 3 4 5

):::::::::::::

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)(τ1τ2τ3τ4)(τ1τ2τ3τ4)

(5 5 5 52 3 4 5

):::::::::::::

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)(τ1τ2τ3τ4)(τ1τ2τ3τ4)

(5 5 5 53 4 5 5

):::::::::::::

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)(τ1τ2τ3τ4)(τ1τ2τ3τ4)

(5 5 5 53 4 5 5

):::::::::::::

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)(τ1τ2τ3τ4)(τ1τ2τ3τ4)

(5 5 5 54 5 5 5

):::::::::::::

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)(τ1τ2τ3τ4)(τ1τ2τ3τ4)

(5 5 5 54 5 5 5

):::::::::::::

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)(τ1τ2τ3τ4)(τ1τ2τ3τ4)

(5 5 5 55 5 5 5

):::::::::::::

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)(τ1τ2τ3τ4)(τ1τ2τ3τ4)

(5 5 5 55 5 5 5

):::::::::::::

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)(τ1τ2τ3τ4)(τ1τ2τ3τ4)Σ

14

::

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)(τ1τ2τ3τ4)(τ1τ2τ3τ4)

Σ14

. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)::::::::::

(τ1τ2τ3τ4)(τ1τ2τ3τ4)

Σ14Yukari Funakoshi Unknotting operations for fibered knots and pseudo-fiber surfaces
. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)::::::::::


. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)::::::::::


. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)::::::::::


. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)::::::::::


. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)::::::::::


. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)::::::::::


. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)::::::::::


. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)::::::::::


. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)::::::::::


. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)::::::::::


. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)::::::::::


. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)::::::::::


. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)::::::::::


. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)::::::::::


. . . . . .



(21

)·(2 31 2

)·(2 3 41 2 3

)::::::::::


. . . . . .



(21

)·(2 31 2

)· (τ1τ2τ3τ4)(τ1τ2τ3τ4)

(2 3 41 2 3

)::::::::::

. . . . . .



(21

)·(2 31 2

)· (τ1τ2τ3τ4)(τ1τ2τ3τ4)

(2 3 41 2 3

)::::::::::

. . . . . .



(21

)·(2 31 2

)· (τ1τ2τ3τ4)

(3 4 52 3 4

)::::::::::

(τ1τ2τ3τ4)

. . . . . .



(21

)·(2 31 2

)· (τ1τ2τ3τ4)

(3 4 52 3 4

)::::::::::

(τ1τ2τ3τ4)

. . . . . .



(21

)·(2 31 2

)· (τ1τ2τ3τ4)

(3 4 52 3 4

)::::::::::

(τ1τ2τ3τ4)

. . . . . .



(21

)·(2 31 2

)· (τ1τ2τ3τ4)

(3 4 53 4 5

)::::::::::

(τ1τ2τ3τ4)

. . . . . .



(21

)·(2 31 2

)· (τ1τ2τ3τ4)(τ1τ2τ3τ4)

(3 4 53 4 5

)::::::::::

. . . . . .



(21

)·(2 31 2

)· (τ1τ2τ3τ4)(τ1τ2τ3τ4)

(3 4 53 4 5

)::::::::::

. . . . . .



(21

)·(2 31 2

)· (τ1τ2τ3τ4)(τ1τ2τ3τ4)

(3 4 54 5 5

)::::::::::

. . . . . .



(21

)·(2 31 2

)· (τ1τ2τ3τ4)(τ1τ2τ3τ4)

(3 4 54 5 5

)::::::::::

. . . . . .



(21

)·(2 31 2

)· (τ1τ2τ3τ4)(τ1τ2τ3τ4)

(3 4 55 5 5

)::::::::::

. . . . . .



(21

)·(2 31 2

)· (τ1τ2τ3τ4)(τ1τ2τ3τ4)

(3 4 55 5 5

)::::::::::

. . . . . .



(21

)·(2 31 2

)· (τ1τ2τ3τ4)(τ1τ2τ3τ4)

(3 4 55 5 5

)::::::::::

. . . . . .



(21

)·(2 31 2

)· (τ1τ2τ3τ4)(τ1τ2τ3τ4)

(3 4 55 5 5

)::::::::::

. . . . . .



(21

)·(2 31 2

)· (τ1τ2τ3τ4)(τ1τ2τ3τ4)

(4 5 55 5 5

)::::::::::

. . . . . .



(21

)·(2 31 2

)· (τ1τ2τ3τ4)(τ1τ2τ3τ4)

(4 5 55 5 5

)::::::::::

. . . . . .



(21

)·(2 31 2

)· (τ1τ2τ3τ4)(τ1τ2τ3τ4)

(5 5 55 5 5

)::::::::::

. . . . . .



(21

)·(2 31 2

)· (τ1τ2τ3τ4)(τ1τ2τ3τ4)

(5 5 55 5 5

)::::::::::

. . . . . .



(21

)·(2 31 2

)· (τ1τ2τ3τ4)(τ1τ2τ3τ4)Σ13

::

Σ14

. . . . . .



(21

)·(2 31 2

)· (τ1τ2τ3τ4)(τ1τ2τ3τ4)

Σ14, Σ13

. . . . . .



(21

)·(2 31 2

):::::::

· (τ1τ2τ3τ4)(τ1τ2τ3τ4)

Σ14, Σ13

. . . . . .



(21

)· (τ1τ2τ3τ4)(τ1τ2τ3τ4)

(2 31 2

):::::::

Σ14, Σ13

. . . . . .



(21

)· (τ1τ2τ3τ4)

(3 42 3

):::::::

(τ1τ2τ3τ4)

Σ14, Σ13

. . . . . .



(21

)· (τ1τ2τ3τ4)

(3 42 3

):::::::

(τ1τ2τ3τ4)

Σ14, Σ13

. . . . . .



(21

)·(4 53 4

):::::::


Σ14, Σ13

. . . . . .



(21

)· (τ1τ2τ3τ4)(τ1τ2τ3τ4)

(4 53 4

):::::::

Σ14, Σ13

. . . . . .



(21

)· (τ1τ2τ3τ4)(τ1τ2τ3τ4)

(4 53 4

):::::::

Σ14, Σ13

. . . . . .



(21

)· (τ1τ2τ3τ4)(τ1τ2τ3τ4)

(4 53 4

):::::::

Σ14, Σ13

. . . . . .



(21

)· (τ1τ2τ3τ4)(τ1τ2τ3τ4)

(4 54 5

):::::::

Σ14, Σ13

. . . . . .



(21

)· (τ1τ2τ3τ4)(τ1τ2τ3τ4)

(4 54 5

):::::::

Σ14, Σ13

. . . . . .



(21

)· (τ1τ2τ3τ4)(τ1τ2τ3τ4)

(4 55 5

):::::::

Σ14, Σ13

. . . . . .



(21

)· (τ1τ2τ3τ4)(τ1τ2τ3τ4)

(4 55 5

):::::::

Σ14, Σ13

. . . . . .



(21

)· (τ1τ2τ3τ4)(τ1τ2τ3τ4)

(4 55 5

):::::::

Σ14, Σ13

. . . . . .



(21

)· (τ1τ2τ3τ4)(τ1τ2τ3τ4)

(4 55 5

):::::::

Σ14, Σ13

. . . . . .



(21

)· (τ1τ2τ3τ4)(τ1τ2τ3τ4)

(5 55 5

):::::::

Σ14, Σ13

. . . . . .



(21

)· (τ1τ2τ3τ4)(τ1τ2τ3τ4)

(5 55 5

):::::::

Σ14, Σ13

. . . . . .



(21

)· (τ1τ2τ3τ4)(τ1τ2τ3τ4)Σ12

::

Σ14, Σ13

. . . . . .



(21

)· (τ1τ2τ3τ4)(τ1τ2τ3τ4)

Σ14, Σ13, Σ

12

. . . . . .




(21

)Σ14, Σ

13, Σ

12

. . . . . .



(τ1τ2τ3τ4)

(32

)(τ1τ2τ3τ4)

Σ14, Σ13, Σ

12

. . . . . .



(43

)(τ1τ2τ3τ4)(τ1τ2τ3τ4)

Σ14, Σ13, Σ

12

. . . . . .




(43

)Σ14, Σ

13, Σ

12

. . . . . .



(τ1τ2τ3τ4)

(54

)(τ1τ2τ3τ4)

Σ14, Σ13, Σ

12

. . . . . .



(τ1τ2τ3τ4)

(54

)(τ1τ2τ3τ4)

Σ14, Σ13, Σ

12

. . . . . .



(τ1τ2τ3τ4)

(55

)(τ1τ2τ3τ4)

Σ14, Σ13, Σ

12

. . . . . .



(55

)(τ1τ2τ3τ4)(τ1τ2τ3τ4)

Σ14, Σ13, Σ

12

. . . . . .



(55

)::::


Σ14, Σ13, Σ

12

. . . . . .



Σ11::


Σ14, Σ13, Σ

12

. . . . . .



(τ1τ2τ3τ4)(τ1τ2τ3τ4)Σ14, Σ

13, Σ

12, Σ

11


Introduction and PreliminariesPre-fiber surfacepseudo-fiber surface of level n

Unknotting operations for bered knots and pseudo- ber surfaces€¦ · Yukari Funakoshi Unknotting...

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Transcript of Unknotting operations for bered knots and pseudo- ber surfaces€¦ · Yukari Funakoshi Unknotting...