Introduction Warping polynomial Span of warping polynomial Span and dealternating number Ayaka Shimizu (Osaka City University) The span of the warping polynomial of a knot diagram

Introduction Warping polynomial Span of warping polynomial Span and dealternating number

.

§ 0. Introduction

D: an oriented knot diagram c(D): the crossing number of D d(D): the warping degree of D cd(D) = (c(D) , d(D)): the

complexity of D

c(D)

cd(D) d(D) d(D ), d(D ), _{a b}

Introduction Warping polynomial Span of warping polynomial Span and dealternating number

.

§ 0. Introduction

D' D

cd(D)=(8,2) cd(D')=(8,2)

c(D)

cd(D) d(D) d(D ), d(D ), _{a b}

Introduction Warping polynomial Span of warping polynomial Span and dealternating number

.

§ 0. Introduction

D'

D 4

5 4 5 4

3 4

5 6 7

6 5

4 3 3 2

4 5 4 3 2

3 2

3 4 5

6 7 6 5 5 6

cd(D)=(8,2) cd(D')=(8,2)

c(D)

cd(D) d(D) d(D ), d(D ), _{a b}

Introduction Warping polynomial Span of warping polynomial Span and dealternating number

.

§ 0. Introduction

D'

D 4

5 4 5 4

3 4

5 6 7

6 5

4 3 2 3

4 5 4 3 2

3 2

3 4 5

6 7

6 5 5 6

D

2 3 4 5 6 7

2 3 4 5 6 7

W (t)=t +3t +5t +4t +t +t

W D(t): the warping polynomial of D W D'(t)=2t +3t +3t +4t +3t +t

c(D)

cd(D) d(D) d(D ), d(D ), _{a b}

W (t)D

Introduction Warping polynomial Span of warping polynomial Span and dealternating number

.

Contents

§ 1. Warping polynomial

§ 2. Span of the warping polynomial

§ 3. Span and dealternating number

Introduction Warping polynomial Span of warping polynomial Span and dealternating number

warping degree of Db warping degree of D warping polynomial properties

.

§ 1. Warping polynomial

§ 1.1. Warping degree of D

§ 1.2. Warping degree of D

§ 1.3. Warping polynomial

§ 1.4. Properties of the warping polynomial

Introduction Warping polynomial Span of warping polynomial Span and dealternating number

warping degree of Db warping degree of D warping polynomial properties

.

§ 1.1. Warping degree of D

D: an oriented knot diagram b: a base point of D

.

.

. . .

. .

A crossing point p of D is a warping crossing point of D

if we meet the point first at the under-crossing when we go along D by starting from b.

D

D b

b

p q

warping non-warping

Introduction Warping polynomial Span of warping polynomial Span and dealternating number

warping degree of Db warping degree of D warping polynomial properties

.

§ 1.1. Warping degree of Db

. .

. . .

.

.

the warping degree of D

d(D

) = ♯{ warping crossing points of D

}

D

d(D )=1 b

b

d(D _c)=0 D

c

Introduction Warping polynomial Span of warping polynomial Span and dealternating number

warping degree of Db warping degree of D warping polynomial properties

.

§ 1.2. Warping degree of D

. .

. . .

.

.

the warping degree of D d(D) =

d(D

)

D

d(D)=0 d(-D)=1 -D

(Warping degree depends on the orientation.)

Introduction Warping polynomial Span of warping polynomial Span and dealternating number

warping degree of Db warping degree of D warping polynomial properties

.

§ 1.3. Warping polynomial

. .

. . .

.

.

Warping degree labeling for D is a labeling s.t.

every edge e has the value d(D

), where b ∈ e.

F D

1 0 1 1 2 1

0 2 0

1 2

3

2 1

E

Introduction Warping polynomial Span of warping polynomial Span and dealternating number

warping degree of Db warping degree of D warping polynomial properties

.

§ 1.3. Warping polynomial

.

.

. . .

.

.

Lemma.

i-1

i+1 i

i

F D

1 0 1 1 2 1

0 2 0

1 2

3

2 1

E

Introduction Warping polynomial Span of warping polynomial Span and dealternating number

warping degree of Db warping degree of D warping polynomial properties

.

§ 1.3. Warping polynomial

D: an oriented knot diagram

.

.

.

.

The warping polynomial W

(t) of D is W

(t) = ∑

e

t

,

where i(e) is the value of an edge e w.r.t. warping degree labeling.

Ayaka Shimizu (Osaka City University) The span of the warping polynomial of a knot diagram

Introduction Warping polynomial Span of warping polynomial Span and dealternating number

warping degree of Db warping degree of D warping polynomial properties

.

§ 1.3. Warping polynomial

F D

1 0 1

D

2 2 3

E F

1 2 1

0 2 0

1 2

3

2 1

E

W (t)=2t+2t

W (t)=2+2t W (t)=1+2t+2t +t

Example.

Introduction Warping polynomial Span of warping polynomial Span and dealternating number

warping degree of Db warping degree of D warping polynomial properties

.

§ 1.4. Properties of the warping polynomial D: an oriented knot diagram with c(D) ≥ 1

− D: D with orientation reversed D

: the mirror image of D

.

.

.

.

• W

(t) = 1

• mindegW

(t) = d(D)

• W

(1) = 2c(D)

• W

( − 1) = 0

• W

(t) = W

(t) = t

W

(t

)

Ayaka Shimizu (Osaka City University) The span of the warping polynomial of a knot diagram

Introduction Warping polynomial Span of warping polynomial Span and dealternating number

span crossing number span 1–3

.

§ 2.1. Span of the warping polynomial

.

. . .

.

.

the span of f (t)

span f (t) = maxdeg f (t) − mindeg f (t)

.

.

.

.

Proposition.

• spanW

(t) = c(D) − (d(D) + d( − D)) .

• ∀ n ≥ 0, ∃ D s.t. spanW

(t) = n.

Ayaka Shimizu (Osaka City University) The span of the warping polynomial of a knot diagram

Introduction Warping polynomial Span of warping polynomial Span and dealternating number

span crossing number span 1–3

.

§ 2.2. Span and crossing number D: a knot diagram with c(D) ≥ 1

.

.

. . .

.

.

Proposition.

spanW

(t) ≤ c(D).

“ = ” ⇔ D is a one-bridge diagram.

D

2 3

4 0

1

Introduction Warping polynomial Span of warping polynomial Span and dealternating number

span crossing number span 1–3

.

§ 2.3. Warping polynomials with span 1–3

.

.

.

.

Theorem.

( i ) f (t) is a warping polynomial with span f (t) = 1

⇔ f (t) = ct

+ ct

, where 1 ≤ c, 1 ≤ d ≤ c − 1.

(ii) f (t) is a warping polynomial with span f (t) = 2

⇔ f (t) = at

+ ct

+ (c − a)t

,

where 2 ≤ c, 1 ≤ a ≤ c − 1, 0 ≤ d ≤ c − 2.

(iii) f (t) is a warping polynomial with span f (t) = 3

⇔ f (t) = at

+ bt

+ (c − a)t

+ (c − b)t

, where 3 ≤ c, 1 ≤ a < b ≤ c − 1, 0 ≤ d ≤ c − 3.

Ayaka Shimizu (Osaka City University) The span of the warping polynomial of a knot diagram

Introduction Warping polynomial Span of warping polynomial Span and dealternating number

dealternating number alternating diagram almost alternating diagram

.

§ 3.1. Span and dealternating number D: a knot diagram

.

.

. . .

.

.

The dealternating number dalt(D) of D is the

minimal number of crossing changes which turn the diagram into an alternating diagram.

.

.

.

.

Proposition.

dalt(D) ≥ spanW

(t) − 1

2 .

Ayaka Shimizu (Osaka City University) The span of the warping polynomial of a knot diagram

Introduction Warping polynomial Span of warping polynomial Span and dealternating number

dealternating number alternating diagram almost alternating diagram

.

§ 3.2. Span and alternating diagram D: a knot diagram with c(D) ≥ 1

.

.

. . .

. .

Theorem [S. 2008].

d(D) + d( − D) + 1 ≤ c(D).

“ = ” ⇔ D is an alternating diagram.

. .

.

.

Corollary.

spanW

(t) = 1 ⇔ D is an alternating diagram.

Ayaka Shimizu (Osaka City University) The span of the warping polynomial of a knot diagram

Introduction Warping polynomial Span of warping polynomial Span and dealternating number

dealternating number alternating diagram almost alternating diagram

.

§ 3.3. Span and almost alternating diagram

.

.

.. .

.

.

Proposition.

If a knot diagram D is analmost alternatingdiagram, then spanW_D(t) is two or three. Furthermore,

( i )if D is obtained from an alternating diagram by a Reidemeister move I, thenspanW_D(t)= 2.

(ii)Otherwise,spanW_D(t) =3.

(i) D (ii) E

2 3

W (t)=1+2t+2t +tE

2 3

W (t)=t+4t +3t D

Introduction Warping polynomial Span of warping polynomial Span and dealternating number

dealternating number alternating diagram almost alternating diagram

.

§ 3.3. Span and almost alternating diagram

.

.

.

.

Lemma of (i).

i i+1

D D'

i i

i

D' D

W ( t ) =W ( t ) +t ( 1+t ) .

i i

D D''

i+1 i+1

i

D'' D

W ( t ) =tW ( t ) +t ( 1+t ) .

Ayaka Shimizu (Osaka City University) The span of the warping polynomial of a knot diagram

Introduction Warping polynomial Span of warping polynomial Span and dealternating number

dealternating number alternating diagram almost alternating diagram

.

§ 3.3. Span and almost alternating diagram

.

.

.

.

Lemma of (ii).

D-

D+

, ,

W

(t) + W

(t) = (1 + t)W

(t).

1

1 1

2

2 2

D+

0

2 2

3

1 1

D-

0

1 1

2

1 1

D0