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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

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

c(D) cd(D) d(D) d(D ), d(D ),

a b

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' 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 6 5

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

c(D) cd(D) d(D) d(D ), d(D ),

a b

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' 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 6 5

D 2 3 4 5 6 7 2 3 4 5 6 7

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

D

W (t): the warping polynomial of D

D'

W (t)=2t +3t +3t +4t +3t +t

c(D) cd(D) d(D) d(D ), d(D ),

a b

D

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

Contents §1. Warping polynomial §2. Span of the warping polynomial §3. 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 warping degree of Db warping degree of D warping polynomial properties

§1. Warping polynomial §1.1. Warping degree of Db §1.2. Warping degree of D §1.3. Warping polynomial §1.4. Properties of the warping polynomial

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

§1.1. Warping degree of Db D: an oriented knot diagram b: a base point of D A crossing point p of D is a warping crossing point

• f Db 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

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

§1.1. Warping degree of Db the warping degree of Db d(Db) = ♯{ warping crossing points of Db}

D d(D )=1 b

b

d(D )=0

c

D c

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

§1.2. Warping degree of D the warping degree of D d(D) = min

b d(Db)

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

• D

(Warping degree depends on the orientation.)

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 Warping degree labeling for D is a labeling s.t. every edge e has the value d(Db), where b ∈ e.

F D 1 1 1 2 1 2 1 2 3 2 1 E

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 Lemma.

i-1 i+1 i i

F D 1 1 1 2 1 2 1 2 3 2 1 E

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 D: an oriented knot diagram The warping polynomial WD(t) of D is WD(t) = ∑

e ti(e),

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 1

D 2 2 3 E F

1 2 1 2 1 2 3 2 1 E

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

Example.

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

§1.4. 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

• W⃝(t) = 1
• mindegWD(t) = d(D)
• WD(1) = 2c(D)
• WD(−1) = 0
• W−D(t) = WD∗(t) = tc(D)WD(t−1)

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.

• spanWD(t) = c(D) − (d(D) + d(−D)).
• ∀n ≥ 0, ∃D s.t. spanWD(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. spanWD(t) ≤ c(D). “=” ⇔ D is a one-bridge diagram. D 2 3 4 1

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

§2.3. Warping polynomials with span 1–3 Theorem. (i) f(t) is a warping polynomial with span f(t) = 1 ⇔ f(t) = ctd + ctd+1, where 1 ≤ c, 1 ≤ d ≤ c − 1. (ii) f(t) is a warping polynomial with span f(t) = 2 ⇔ f(t) = atd + ctd+1 + (c − a)td+2, 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) = atd + btd+1 + (c − a)td+2 + (c − b)td+3, 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) ≥ spanWD(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. spanWD(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 an almost alternating diagram, then spanWD(t) is two or three. Furthermore, (i)if D is obtained from an alternating diagram by a Reidemeister move I, then spanWD(t) = 2. (ii)Otherwise, spanWD(t) = 3.

D (i) (ii) E

2 3 E

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

2 3 D

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

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 (i).

i+1 i

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+ D0

, ,

WD+(t) + WD−(t) = (1 + t)WD0(t).

1 1 1 2 2 2

D+

2 2 3 1 1

D-

1 1 2 1 1

D0

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