Gromov hyperbolicity of a variation of the Gordian complex

結び目の数学II (_{早稲田大学} 14_号館 B101_教室) 2009/12/24 (_木), 16:40–17:10.

Gromov hyperbolicity of a variation of the

Gordian complex

In Dae Jong (Osaka City University) joint work with

Kazuhiro Ichihara (Nara University of Education)

§ 1. Introduction

K := { knots in the 3-sphere } . K , K ^′ ∈ K ^．

K ↔ ^x K ^′ ⇔ K can be deformed into K ^′ by once.

Gordian distance d ^x ( · , · )

¶ ³

d ^x ( K, K ^′ ) := min { n | K = K ₀ ↔ · · · ^x ↔ ^x K _n = K ^′ } .

µ ´

Gordian graph G ^x

¶ ³

• { vertices } = K .

• ∃ an edge between K and K ^′ ⇔ d ^x ( K, K ^′ ) = 1.

µ ´

Remark 1.1

¶ ³

G ^x = the 1–skelton of the Gordian complex

introduced by [Hirasawa-Uchida ’02].

µ ´

§ 1. Introduction

K := { knots in the 3-sphere } . K , K ^′ ∈ K ^．

K ↔ ^x K ^′ ⇔ K can be deformed into K ^′ by once.

Gordian distance d ^x ( · , · )

¶ ³

d ^x ( K, K ^′ ) := min { n | K = K ₀ ↔ · · · ^x ↔ ^x K _n = K ^′ } .

µ ´

Gordian graph G ^x

¶ ³

• { vertices } = K .

• ∃ an edge between K and K ^′ ⇔ d ^x ( K, K ^′ ) = 1.

µ ´

Remark 1.1

¶ ³

G ^x = the 1–skelton of the Gordian complex

introduced by [Hirasawa-Uchida ’02].

µ ´

-14-a

Gordian graph G ^x

¶ ³

• { vertices } = K .

• ∃ an edge between K and K ^′ ⇔ d ^x ( K, K ^′ ) = 1.

µ ´

Remark 1.1

¶ ³

G ^x = the 1–skelton of the Gordian complex

introduced by [Hirasawa-Uchida ’02].

µ ´

Gordian graph G ^x

¶ ³

• { vertices } = K .

• ∃ an edge between K and K ^′ ⇔ d ^x ( K, K ^′ ) = 1.

µ ´

Remark 1.1

¶ ³

G ^x = the 1–skelton of the Gordian complex

introduced by [Hirasawa-Uchida ’02].

µ ´

-14-c

Gordian graph G ^x

¶ ³

• { vertices } = K .

• ∃ an edge between K and K ^′ ⇔ d ^x ( K, K ^′ ) = 1.

µ ´

Remark 1.1

¶ ³

G ^x = the 1–skelton of the Gordian complex

introduced by [Hirasawa-Uchida ’02].

µ ´

Gordian graph G ^x

¶ ³

• { vertices } = K .

• ∃ an edge between K and K ^′ ⇔ d ^x ( K, K ^′ ) = 1.

µ ´

Remark 1.1

¶ ³

G ^x = the 1–skelton of the Gordian complex

introduced by [Hirasawa-Uchida ’02].

µ ´

-14-e

Gordian graph G ^x

¶ ³

• { vertices } = K .

• ∃ an edge between K and K ^′ ⇔ d ^x ( K, K ^′ ) = 1.

µ ´

Proposition 1.2 [Gambaudo-Ghys ’05].

¶ ³

The Gordian graph G ^x is not Gromov hyperbolic.

(We introduce the Gromov hyperbolicity later. )

µ ´

λ : a local move on knots _．

K ↔ ^λ K ^′ ⇔ K can be deformed into K ^′ by λ once.

λ -Gordian distance d ^λ ( · , · )

¶ ³

d ^λ ( K, K ^′ ) := min { n | K = K ₀ ↔ · · · ^λ ↔ ^λ K _n = K ^′ } .

µ ´

λ -Gordian graph G ^λ

¶ ³

• { vertices } = K .

• ∃ an edge between K and K ^′ ⇔ d ^λ ( K, K ^′ ) = 1.

µ ´

Problem 1.3.

¶ ³

For a given local move λ ,

detect whether G ^λ is Gromov hyperbolic or not.

µ ´

-13

λ : a local move on knots _．

K ↔ ^λ K ^′ ⇔ K can be deformed into K ^′ by λ once.

λ -Gordian distance d ^λ ( · , · )

¶ ³

d ^λ ( K, K ^′ ) := min { n | K = K ₀ ↔ · · · ^λ ↔ ^λ K _n = K ^′ } .

µ ´

λ -Gordian graph G ^λ

¶ ³

• { vertices } = K .

• ∃ an edge between K and K ^′ ⇔ d ^λ ( K, K ^′ ) = 1.

µ ´

Problem 1.3.

¶ ³

For a given local move λ ,

detect whether G ^λ is Gromov hyperbolic or not.

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§ 2. Gromov hyperbolicity

Assumption on graphs

¶ ³

Graphs are connected & each edge has length 1.

µ ´

Γ : a graph. N ( γ, ε ) : ε -nbd. of γ ⊂ Γ.

δ -thin

¶ ³

T : a triangle with sides s ₁ , s ₂ , s ₃ in Γ.

T is δ -thin ⇔ s _i ⊂ N ( s _j ∪ s _k , δ ) for different i, j, k .

µ ´

-12

§ 2. Gromov hyperbolicity

Assumption on graphs

¶ ³

Graphs are connected & each edge has length 1.

µ ´

Γ : a graph. N ( γ, ε ) : ε -nbd. of γ ⊂ Γ.

δ -thin

¶ ³

T : a triangle with sides s ₁ , s ₂ , s ₃ in Γ.

T is δ -thin ⇔ s _i ⊂ N ( s _j ∪ s _k , δ ) for different i, j, k .

µ ´

δ s

s

s

s

⊂ N ( s

∪ s

, δ ).

Hyperbolicity

¶ ³

Γ is δ -hyperbolic (or Gromov hyperbolic) ⇔

any geodesic triangle in Γ is δ -thin for δ > 0.

µ ´

(A geodesic triangle ⇔ each side is a geodesic.) Example 2.1

¶ ³

• Any tree is 0-hyperbolic.

• ^{R 2} is not Gromov hyperbolic.

• ^{H 2} is ^{log 3}

2 -hyperbolic.

• A graph with finite diameter, r , is r -hyperbolic.

µ ´

Example 2.2 [Masur-Minsky ’99]

¶ ³

For a surface S , the curve complex C ( S ) is Gromov hyperbolic. The constant δ depends on S .

µ ´

-11

Hyperbolicity

¶ ³

Γ is δ -hyperbolic (or Gromov hyperbolic) ⇔

any geodesic triangle in Γ is δ -thin for δ > 0.

µ ´

(A geodesic triangle ⇔ each side is a geodesic.) Example 2.1

¶ ³

• Any tree is 0-hyperbolic.

• ^{R 2} is not Gromov hyperbolic.

• ^{H 2} is ^{log 3}

2 -hyperbolic.

• A graph with finite diameter r is r -hyperbolic.

µ ´

Example 2.2 [Masur-Minsky ’99]

¶ ³

For a surface S , the curve complex C ( S ) is Gromov hyperbolic. The constant δ depends on S .

µ ´

Hyperbolicity

¶ ³

Γ is δ -hyperbolic (or Gromov hyperbolic) ⇔

any geodesic triangle in Γ is δ -thin for δ > 0.

µ ´

(A geodesic triangle ⇔ each side is a geodesic.) Example 2.1

¶ ³

• Any tree is 0-hyperbolic.

• ^{R 2} is not Gromov hyperbolic.

• ^{H 2} is ^{log 3}

2 -hyperbolic.

• A graph with finite diameter r is r -hyperbolic.

µ ´

Example 2.2 [Masur-Minsky ’99]

¶ ³

For a surface S , the curve complex C ( S ) is Gromov hyperbolic. The constant δ depends on S .

µ ´

-11-b

§ 3. ( ι, λ )-Gordian graph

ι : a knot invariant.

K ∼ ^ι K ^′ : ⇔ ι ( K ) = ι ( K ^′ ) for K, K ^′ ∈ K .

[ K ] _ι : the equivalence class of K w.r.t. ∼ ^ι . K ^ι := { [ K ] _ι | K ∈ K }

( ι, λ )-Gordian graph G ^λ _ι

¶ ³

• { vertices } = K ^ι .

• ∃ an edge between [ K ] _ι and [ K ^′ ] _ι ⇔

∃ J ∈ [ K ] _ι , ∃ J ^′ ∈ [ K ^′ ] _ι s.t. d ^λ ( J, J ^′ ) = 1.

µ ´

d ^λ _ι : the metric on G ^λ _ι .

§ 3. ( ι, λ )-Gordian graph

ι : a knot invariant.

K ∼ ^ι K ^′ : ⇔ ι ( K ) = ι ( K ^′ ) for K, K ^′ ∈ K .

[ K ] _ι : the equivalence class of K w.r.t. ∼ ^ι . K ^ι := { [ K ] _ι | K ∈ K }

( ι, λ )-Gordian graph G ^λ _ι

¶ ³

• { vertices } = K ^ι .

• ∃ an edge between [ K ] _ι and [ K ^′ ] _ι ⇔

∃ J ∈ [ K ] _ι , ∃ J ^′ ∈ [ K ^′ ] _ι s.t. d ^λ ( J, J ^′ ) = 1.

µ ´

d ^λ _ι : the metric on G ^λ _ι .

-10-a

∇ _K : the Conway polynomial of a knot K . ( ∇ _K = 1 + a ₂ z ² + · · · + a _{2 n} z ^{2 n} .)

Proposition 3.1.

¶ ³

The ( ∇ , x)-Gordian graph G ^x _∇ has diameter 2.

µ ´

Proof. Any [ K ] _∇ contains a knot with unknotting number 1 [Kondo], [Sakai] . Thus, diam G ^x _∇ ≤ 2. On the other hand, d ^x _∇ ([3 ₁ ] _∇ , [4 ₁ ] _∇ ) = 2 [Kawauchi] . _¤

Corollary 3.2.

The graph G ^x _∇ is Gromov hyperbolic.

Proof. Any graph with finite diameter is

Gromov hyperbolic. _¤

∇ _K : the Conway polynomial of a knot K . ( ∇ _K = 1 + a ₂ z ² + · · · + a _{2 n} z ^{2 n} .)

Proposition 3.1.

¶ ³

The ( ∇ , x)-Gordian graph G ^x _∇ has diameter 2.

µ ´

Proof. Any [ K ] _∇ contains a knot with unknotting number 1 [Kondo], [Sakai] . Thus, diam G ^x _∇ ≤ 2. On the other hand, d ^x _∇ ([3 ₁ ] _∇ , [4 ₁ ] _∇ ) = 2 [Kawauchi] . _¤

Corollary 3.2.

The graph G ^x _∇ is Gromov hyperbolic.

Proof. Any graph with finite diameter is

Gromov hyperbolic. _¤

-9-a

∇ _K : the Conway polynomial of a knot K . ( ∇ _K = 1 + a ₂ z ² + · · · + a _{2 n} z ^{2 n} .)

Proposition 3.1.

¶ ³

The ( ∇ , x)-Gordian graph G ^x _∇ has diameter 2.

µ ´

Proof. Any [ K ] _∇ contains a knot with unknotting number 1 [Kondo], [Sakai] . Thus, diam G ^x _∇ ≤ 2. On the other hand, d ^x _∇ ([3 ₁ ] _∇ , [4 ₁ ] _∇ ) = 2 [Kawauchi] . _¤

Corollary 3.2.

The graph G ^x _∇ is Gromov hyperbolic.

Proof. Any graph with finite diameter is

Gromov hyperbolic. _¤

∇ _K : the Conway polynomial of a knot K . ( ∇ _K = 1 + a ₂ z ² + · · · + a _{2 n} z ^{2 n} .)

Proposition 3.1.

¶ ³

The ( ∇ , x)-Gordian graph G ^x _∇ has diameter 2.

µ ´

Proof. Any [ K ] _∇ contains a knot with unknotting number 1 [Kondo], [Sakai] . Thus, diam G ^x _∇ ≤ 2. On the other hand, d ^x _∇ ([3 ₁ ] _∇ , [4 ₁ ] _∇ ) = 2 [Kawauchi] . _¤

Corollary 3.2.

The graph G ^x _∇ is Gromov hyperbolic.

Proof. Any graph with finite diameter is

Gromov hyperbolic. _¤

-9-c

Delta-move ∆ [Matveev] [Murakami-Nakanishi]

¶ ³

µ ´

We focus on the graph G ^∆ _∇ with metric d ^∆ _∇ . We call d ^∆ _∇ the ( ∇ , ∆)-Gordian distance.

Main Theorem [Ichihara-J.].

The graph G ^∆ _∇ is Gromov hyperbolic.

Remark 2.3.

¶ ³

The diameter of G ^∆ _∇ is infinite.

µ ´

Delta-move ∆ [Matveev] [Murakami-Nakanishi]

¶ ³

µ ´

We focus on the graph G ^∆ _∇ with metric d ^∆ _∇ . We call d ^∆ _∇ the ( ∇ , ∆)-Gordian distance.

Main Theorem [Ichihara-J.].

The graph G ^∆ _∇ is Gromov hyperbolic.

Remark 3.3.

¶ ³

The diameter of G ^∆ _∇ is infinite.

µ ´

-8-a

A local picture of G ^∆ _∇

a

( m ≥ 4)

a

[ K ]

§ 4. ( ∇ , ∆)-Gordian distance d ^∆ _∇

Lemma 4.1 [Okada ’90].

¶ ³

Let K and K ^′ be knots with d ^∆ ( K, K ^′ ) = 1.

Then we have a ₂ ( K ) − a ₂ ( K ^′ ) = ± 1 .

µ ´

Notation: [ K ] = [ K ] _∇ .

Lemma 4.2.

¶ ³

For any [ K ] , [ K ^′ ], we have the following.

• d ^∆ _∇ ([ K ] , [ K ^′ ]) ≥ | a ₂ ( K ) − a ₂ ( K ^′ ) | .

• d ^∆ _∇ ([ K ] , [ K ^′ ]) ≡ | a ₂ ( K ) − a ₂ ( K ^′ ) | mod 2.

µ ´

Recall: For K ₁ , K ₂ ∈ [ K ], we have ∇ K

= ∇ K

.

-6

§ 4. ( ∇ , ∆)-Gordian distance d ^∆ _∇

Lemma 4.1 [Okada ’90].

¶ ³

Let K and K ^′ be knots with d ^∆ ( K, K ^′ ) = 1.

Then we have a ₂ ( K ) − a ₂ ( K ^′ ) = ± 1 .

µ ´

Notation: [ K ] = [ K ] _∇ .

Lemma 4.2.

¶ ³

For any [ K ] , [ K ^′ ], we have the following.

• d ^∆ _∇ ([ K ] , [ K ^′ ]) ≥ | a ₂ ( K ) − a ₂ ( K ^′ ) | .

• d ^∆ _∇ ([ K ] , [ K ^′ ]) ≡ | a ₂ ( K ) − a ₂ ( K ^′ ) | mod 2.

µ ´

Recall: For K ₁ , K ₂ ∈ [ K ], we have ∇ K

= ∇ K

.

Lemma 4.3.

¶ ³

For [ K ] ̸ = [ K ^′ ] with a ₂ = a ₂ ( K ), a ^′ ₂ = a ₂ ( K ^′ ), (1) a ₂ = a ^′ ₂ ⇒ d ^∆ _∇ ([ K ] , [ K ^′ ]) = 2.

(2) | a ₂ − a ^′ ₂ | ≥ 2 ⇒ d ^∆ _∇ ([ K ] , [ K ^′ ]) = | a ₂ − a ^′ ₂ | . (3) | a ₂ − a ^′ ₂ | = 1, ⇒ d ^∆ _∇ ([ K ] , [ K ^′ ]) = 1 or 3.

µ ´

The proof are achieved by constructing knots which satisfy given conditions. We give the proof later.

… … … … ……

…

…

…

-5

§ 4. Proof of Main Theorem

V _n := { [ K ] ∈ K _∇ | a ₂ ( K ) = n } .

S _n : the subgraph of G ^∆ _∇ induced by V _n ∪ V _{n ± 1} . Lemma 5.1.

¶ ³

For [ K ] ∈ K _∇ with a ₂ ( K ) = n , N ([ K ] , 3) ⊃ S _n .

µ ´

(The proof is immediately obtained by Lemma 4.3.)

a_m (m ≥ 4)

a₂

V_n V_n

−1 V_n+1

S

}

[K]

Main Theorem [Ichihara-J.].

The graph G ^∆ _∇ is Gromov hyperbolic.

Proof of Main Theorem There are several cases.

We only show the theorem for a particular case.

Other cases are shown in a similar way.

-3

T : a geodesic triangle with sides s ₁ , s ₂ , s ₃ . Let

s ₁ = x ₀ x ₁ ∪ x ₁ x ₂ ∪ · · · ∪ x _{p − 1} x _p , s ₂ = y ₀ y ₁ ∪ y ₁ y ₂ ∪ · · · ∪ y _{q − 1} y _q , s ₃ = z ₀ z ₁ ∪ z ₁ z ₂ ∪ · · · ∪ z _{r − 1} z _r ,

where x ₀ , . . . , x _p , y ₀ , . . . , y _q , z ₀ , . . . , z _r ∈ K _∇ with

x ₀ = x = z _r , y ₀ = y = x _p , and z ₀ = z = y _q .

a

a

x

x

x

y

y

y

z

z

z

z

z

z

z

x

y

z

x

y

z

The figure is an example of T ( p = 4, q = 4, r = 8).

To show: T is 3-thin, namely,

N ( s ( x, y ) ∪ s ( y, z ) , 3) ⊃ s ( z, x ), N ( s ( y, z ) ∪ s ( z, x ) , 3) ⊃ s ( x, y ), and N ( s ( z, x ) ∪ s ( x, y ) , 3) ⊃ s ( y, z ).

-1

a

a

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x

y

z

By Lemma 5.1, N ( y _j , 3) ⊃ z _{q − j +1} z _{q − j} , z _{q − j} z _{q − j − 1} for j = 1 , · · · , q − 1. Thus, we have

N ( s ( y, z ) , 3) ⊃ z _q z _{q − 1} ∪ · · · ∪ z ₁ z ₀ .

Therefore we have N ( s ( x, y ) ∪ s ( y, z ) , 3) ⊃ s ( z, x ).

a

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x

y

y

y

z

z

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z

x

y

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x

y

z

By Lemma 5.1, N ( y _j , 3) ⊃ z _{q − j +1} z _{q − j} , z _{q − j} z _{q − j − 1} for j = 1 , · · · , q − 1. Thus, we have

N ( s ( y, z ) , 3) ⊃ z _q z _{q − 1} ∪ · · · ∪ z ₁ z ₀ .

Therefore we have N ( s ( x, y ) ∪ s ( y, z ) , 3) ⊃ s ( z, x ).

-1-b

a

a

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x

x

y

y

y

z

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y

z

Similarly, N ( x _j , 3) ⊃ z _{r − j +1} z _{r − j} , z _{r − j} z _{r − j − 1} for j = 1 , · · · , p − 1. Thus, we have

N ( s ( x, y ) , 3) ⊃ z _r z _{r − 1} ∪ · · · ∪ z _{p +1} z _p .

Therefore we have N ( s ( x, y ) ∪ s ( y, z ) , 3) ⊃ s ( z, x ).

a

a

x

x

x

y

y

y

z

z

z

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z

z

z

x

y

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x

y

z

Similarly, N ( x _j , 3) ⊃ z _{r − j +1} z _{r − j} , z _{r − j} z _{r − j − 1} for j = 1 , · · · , p − 1. Thus, we have

N ( s ( x, y ) , 3) ⊃ z _r z _{r − 1} ∪ · · · ∪ z _{p +1} z _p .

Therefore we have N ( s ( x, y ) ∪ s ( y, z ) , 3) ⊃ s ( z, x ).

-1-d

a

a

x

x

x

y

y

y

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x

y

z

Similarly, N ( x _j , 3) ⊃ z _{r − j +1} z _{r − j} , z _{r − j} z _{r − j − 1} for j = 1 , · · · , p − 1. Thus, we have

N ( s ( x, y ) , 3) ⊃ z _r z _{r − 1} ∪ · · · ∪ z _{p +1} z _p .

Therefore we have N ( s ( x, y ) ∪ s ( y, z ) , 3) ⊃ s ( z, x ).

Remaining two conditions

• N ( s ( y, z ) ∪ s ( z, x ) , 3) ⊃ s ( x, y ) and

• N ( s ( z, x ) ∪ s ( x, y ) , 3) ⊃ s ( y, z )

are shown by the similar argument.

Therefore the geodesic triangle T is 3-thin. _¤

0

Lemma 4.3.

¶ ³

For [ K ] ̸ = [ K ^′ ] with a ₂ = a ₂ ( K ), a ^′ ₂ = a ₂ ( K ^′ ), (1) a ₂ = a ^′ ₂ ⇒ d ^∆ _∇ ([ K ] , [ K ^′ ]) = 2.

(2) | a ₂ − a ^′ ₂ | ≥ 2 ⇒ d ^∆ _∇ ([ K ] , [ K ^′ ]) = | a ₂ − a ^′ ₂ | . (3) | a ₂ − a ^′ ₂ | = 1, ⇒ d ^∆ _∇ ([ K ] , [ K ^′ ]) = 1 or 3.

µ ´

Proof. Let K _m be the twist knot. Then we have

∇ K

= 1 + mz ² , and d ^∆ ( K _{m +1} , K _m ) = 1.

Lemma 4.3.

¶ ³

For [ K ] ̸ = [ K ^′ ] with a ₂ = a ₂ ( K ), a ^′ ₂ = a ₂ ( K ^′ ), (1) a ₂ = a ^′ ₂ ⇒ d ^∆ _∇ ([ K ] , [ K ^′ ]) = 2.

(2) | a ₂ − a ^′ ₂ | ≥ 2 ⇒ d ^∆ _∇ ([ K ] , [ K ^′ ]) = | a ₂ − a ^′ ₂ | . (3) | a ₂ − a ^′ ₂ | = 1, ⇒ d ^∆ _∇ ([ K ] , [ K ^′ ]) = 1 or 3.

µ ´

Proof. Let K _m be the twist knot. Then we have

∇ K

= 1 + mz ² , and d ^∆ ( K _{m +1} , K _m ) = 1.

…

m-full twists

-3-a

Let K _± ( α ₁ , . . . , α _n ) be knots as following.

… … … … ……

…

…

…

α₁

}

α₂

}

α₃

} }

}

}

α_n … … … …

…

…

…

α₁

}

α₂

}

α₃

} }

}

}

α_n

K₊(α₁… α_n) K₋(α₁… α_n)

By [Murakami], [Yamada] , we have

∇ _K

_{( α}

_,...,α

₎ = 1 +

ⁿ

i =1

( − 1) ^{i − 1} α _i z ^{2 i} ,

and d ^∆ ( K _± ( α ₁ , . . . , α _n ) , K _α

_{± 1} ) = 1.

For [ K ] ̸ = [ K ^′ ] ∈ K _∇ , let

∇ K = 1 +

ⁿ _{i =1} a _{2 i} z ^{2 i} , ∇ _K

= 1 +

^m _{i =1} a ^′ _{2 i} z ^{2 i} , J ₊ = K ₊ ( a ₂ , . . . , ( − 1) ^{n − 1} a _{2 n} ), and

J _± ^′ = K _± ( a ^′ ₂ , . . . , ( − 1) ^{m − 1} a ^′ _{2 m} ).

(1) a ₂ = a ^′ ₂ ⇒ d ^∆ _∇ ([ K ] , [ K ^′ ]) = 2.

• We have d ^∆ _∇ ([ K ] , [ K ^′ ]) ≥ 2 by Lemma 4.2.

• We have d ^∆ _∇ ([ K ] , [ K ^′ ]) ≤ 2 by the sequence

of knots J ₊ , K _a

₊₁ , J ₊ ^′ . ( d ^∆ ( J ₊ , J ₊ ^′ ) ≤ 2, ∇ J

= ∇ K , and ∇ _J

+

= ∇ _K

.)

-1

For [ K ] ̸ = [ K ^′ ] ∈ K _∇ , let

∇ K = 1 +

ⁿ _{i =1} a _{2 i} z ^{2 i} , ∇ _K

= 1 +

^m _{i =1} a ^′ _{2 i} z ^{2 i} , J ₊ = K ₊ ( a ₂ , . . . , ( − 1) ^{n − 1} a _{2 n} ), and

J _± ^′ = K _± ( a ^′ ₂ , . . . , ( − 1) ^{m − 1} a ^′ _{2 m} ).

(1) a ₂ = a ^′ ₂ ⇒ d ^∆ _∇ ([ K ] , [ K ^′ ]) = 2.

• We have d ^∆ _∇ ([ K ] , [ K ^′ ]) ≥ 2 by Lemma 4.2.

• We have d ^∆ _∇ ([ K ] , [ K ^′ ]) ≤ 2 by the sequence

of knots J ₊ , K _a

₊₁ , J ₊ ^′ . ( d ^∆ ( J ₊ , J ₊ ^′ ) ≤ 2, ∇ J

= ∇ K , and ∇ _J

+

= ∇ _K

.)

For [ K ] ̸ = [ K ^′ ] ∈ K _∇ , let

∇ K = 1 +

ⁿ _{i =1} a _{2 i} z ^{2 i} , ∇ _K

= 1 +

^m _{i =1} a ^′ _{2 i} z ^{2 i} , J ₊ = K ₊ ( a ₂ , . . . , ( − 1) ^{n − 1} a _{2 n} ), and

J _± ^′ = K _± ( a ^′ ₂ , . . . , ( − 1) ^{m − 1} a ^′ _{2 m} ).

(1) a ₂ = a ^′ ₂ ⇒ d ^∆ _∇ ([ K ] , [ K ^′ ]) = 2.

• We have d ^∆ _∇ ([ K ] , [ K ^′ ]) ≥ 2 by Lemma 4.2.

• We have d ^∆ _∇ ([ K ] , [ K ^′ ]) ≤ 2 by the sequence

of knots J ₊ , K _a

₊₁ , J ₊ ^′ . ( d ^∆ ( J ₊ , J ₊ ^′ ) ≤ 2, ∇ J

= ∇ K , and ∇ _J

+

= ∇ _K

.)

-1-b

(2) | a ₂ − a ^′ ₂ | ≥ 2 ⇒ d ^∆ _∇ ([ K ] , [ K ^′ ]) = | a ₂ − a ^′ ₂ | . We may assume that a ^′ ₂ ≥ a ₂ + 2.

• We have d ^∆ _∇ ([ K ] , [ K ^′ ]) ≥ a ^′ ₂ − a ₂ by Lemma 4.2.

• We have d ^∆ _∇ ([ K ] , [ K ^′ ]) ≤ a ^′ ₂ − a ₂ by the sequence of knots J ₊ , K _a

2

+1 , . . . , K _a

2

− 1 , J ₋ ^′ . (3) | a ₂ − a ^′ ₂ | = 1, ⇒ d ^∆ _∇ ([ K ] , [ K ^′ ]) = 1 or 3.

We may assume that a ^′ ₂ = a ₂ + 1.

• We have d ^∆ _∇ ([ K ] , [ K ^′ ]) ≡ 1 mod 2 by Lemma 4.2.

• We have d ^∆ _∇ ([ K ] , [ K ^′ ]) ≤ 3 by the sequence

of knots J ₊ , K _a

₊₁ , K _a

, J ₋ ^′ .

¤ (Lemma 4.3)

(2) | a ₂ − a ^′ ₂ | ≥ 2 ⇒ d ^∆ _∇ ([ K ] , [ K ^′ ]) = | a ₂ − a ^′ ₂ | . We may assume that a ^′ ₂ ≥ a ₂ + 2.

• We have d ^∆ _∇ ([ K ] , [ K ^′ ]) ≥ a ^′ ₂ − a ₂ by Lemma 4.2.

• We have d ^∆ _∇ ([ K ] , [ K ^′ ]) ≤ a ^′ ₂ − a ₂ by the sequence of knots J ₊ , K _a

2

+1 , . . . , K _a

2

− 1 , J ₋ ^′ . (3) | a ₂ − a ^′ ₂ | = 1, ⇒ d ^∆ _∇ ([ K ] , [ K ^′ ]) = 1 or 3.

We may assume that a ^′ ₂ = a ₂ + 1.

• We have d ^∆ _∇ ([ K ] , [ K ^′ ]) ≡ 1 mod 2 by Lemma 4.2.

• We have d ^∆ _∇ ([ K ] , [ K ^′ ]) ≤ 3 by the sequence

of knots J ₊ , K _a

₊₁ , K _a

, J ₋ ^′ .

¤ (Lemma 4.3)

0-a