Arc presentations of and links

Hwa Jeong Lee (KAIST)

December 10, 2013 IBS Table of Contents

1 What is a ?

2 Arc presentation and Arc index

3 Arc index of Kanenobu knots (joint with Hideo Takioka)

4 A relation between arc index and Thurston-Bennequin number Table of Contents

1 What is a Knot?

2 Arc presentation and Arc index

3 Arc index of Kanenobu knots (joint with Hideo Takioka)

4 A relation between arc index and Thurston-Bennequin number Table of Contents

1 What is a Knot?

2 Arc presentation and Arc index

3 Arc index of Kanenobu knots (joint with Hideo Takioka)

4 A relation between arc index and Thurston-Bennequin number Table of Contents

1 What is a Knot?

2 Arc presentation and Arc index

3 Arc index of Kanenobu knots (joint with Hideo Takioka)

4 A relation between arc index and Thurston-Bennequin number What is ?

Unknottedcircle Knottedcircle Unknottedcircle

Knot theory in the mathematical sense is the study of closed curves in R3. What is Knot Theory?

Unknottedcircle Knottedcircle Unknotted circle

Knot theory in the mathematical sense is the study of closed curves in R3. What is Knot Theory?

Unknottedcircle Knottedcircle Unknottedcircle

Knot theory in the mathematical sense is the study of closed curves in R3. What is Knot Theory?

Unknottedcircle Knottedcircle Unknotted circle

Knot theory in the mathematical sense is the study of closed curves in R3. What is Knot Theory?

Unknottedcircle Knottedcircle Unknottedcircle

Knot theory in the mathematical sense is the study of closed curves in R3. What is Knot Theory?

Unknottedcircle Knottedcircle Unknottedcircle

Knot theory in the mathematical sense is the study of closed curves in R3. What is a Knot?

• A knot is an embedding of a circle in 3-dimensional Euclidean space, R3. • A is a disjoint union of knots in R3.

Main Problem How to classify knots? What is a Knot?

• A knot is an embedding of a circle in 3-dimensional Euclidean space, R3. • A link is a disjoint union of knots in R3.

Unknot

Main Problem How to classify knots? What is a Knot?

• A knot is an embedding of a circle in 3-dimensional Euclidean space, R3. • A link is a disjoint union of knots in R3.

Unknot

Main Problem How to classify knots? What is a Knot?

• A knot is an embedding of a circle in 3-dimensional Euclidean space, R3. • A link is a disjoint union of knots in R3.

Unknot Trefoil knot

Main Problem How to classify knots? What is a Knot?

• A knot is an embedding of a circle in 3-dimensional Euclidean space, R3. • A link is a disjoint union of knots in R3.

Unknot Trefoil knot

Main Problem How to classify knots? What is a Knot?

• A knot is an embedding of a circle in 3-dimensional Euclidean space, R3. • A link is a disjoint union of knots in R3.

Unknot Trefoil knot Hopf link

Main Problem How to classify knots? Two knots are equivalent

3 Two knots K1 and K2 in R are equivalent (K1 ∼ K2) if there is an orientation 3 3 preserving homeomorphism h : R → R such that h(K1) = K2.

Knots that are equivalent can be regarded as being simply the same (type) knot.

Question 1.

How do we know these are actually same or different knots ? Two knots are equivalent

3 Two knots K1 and K2 in R are equivalent (K1 ∼ K2) if there is an orientation 3 3 preserving homeomorphism h : R → R such that h(K1) = K2.

Knots that are equivalent can be regarded as being simply the same (type) knot.

Question 1.

How do we know these are actually same or different knots ? Two knots are equivalent

3 Two knots K1 and K2 in R are equivalent (K1 ∼ K2) if there is an orientation 3 3 preserving homeomorphism h : R → R such that h(K1) = K2.

Knots that are equivalent can be regarded as being simply the same (type) knot.

Question 1.

∼ , ≁

How do we know these are actually same or different knots ? Two knots are equivalent

3 Two knots K1 and K2 in R are equivalent (K1 ∼ K2) if there is an orientation 3 3 preserving homeomorphism h : R → R such that h(K1) = K2.

Knots that are equivalent can be regarded as being simply the same (type) knot.

Question 1.

∼ , ≁

How do we know these are actually same or different knots ? Knot diagram

• A knot diagram is a regular projection of a knot that has relative height information at each of the double points. underpass • The crossing number, c(K), of a knot K is the minimal number of crossings in any knot diagram for K. overpass

• A knot diagram is alternating if the crossings alternate between over and under as one travels around the knot in a fixed direction. • An is a knot which admits an alternating diagram. Knot diagram

• A knot diagram is a regular projection of a knot that has relative height information at each of the double points. underpass • The crossing number, c(K), of a knot K is the minimal number of crossings in any knot diagram for K. overpass

• A knot diagram is alternating if the crossings alternate between over and under as one travels around the knot in a fixed direction. • An alternating knot is a knot which admits an alternating diagram. Knot diagram

• A knot diagram is a regular projection of a knot that has relative height information at each of the double points. underpass • The crossing number, c(K), of a knot K is the minimal number of crossings in any knot diagram for K. overpass

• A knot diagram is alternating if the crossings alternate between over and under as one travels around the knot in a fixed direction. • An alternating knot is a knot which admits an alternating diagram. Knot diagram

• A knot diagram is a regular projection of a knot that has relative height information at each of the double points. underpass • The crossing number, c(K), of a knot K is the minimal number of crossings in any knot diagram for K. overpass

• A knot diagram is alternating if the crossings alternate between over and under as one travels around the knot in a fixed direction. • An alternating knot is a knot which admits an alternating diagram. Knot diagram

• A knot diagram is a regular projection of a knot that has relative height information at each of the double points. underpass • The crossing number, c(K), of a knot K is the minimal number of crossings in any knot diagram for K. overpass

• A knot diagram is alternating if the crossings alternate between over and under as one travels around the knot in a fixed direction. • An alternating knot is a knot which admits an alternating diagram. Reidemeister Theorem

Reidemeister Theorem Two knot diagrams correspond to the same type knot iff one can be obtained from the other by a finite sequence of Reidemeister moves and plane isotopies. Reidemeister Theorem

Reidemeister Theorem Two knot diagrams correspond to the same type knot iff one can be obtained from the other by a finite sequence of Reidemeister moves and plane isotopies.

I I Reidemeister Theorem

Reidemeister Theorem Two knot diagrams correspond to the same type knot iff one can be obtained from the other by a finite sequence of Reidemeister moves and plane isotopies.

I I

II II Reidemeister Theorem

Reidemeister Theorem Two knot diagrams correspond to the same type knot iff one can be obtained from the other by a finite sequence of Reidemeister moves and plane isotopies.

I I

II II

III Reidemeister Theorem

Reidemeister Theorem Two knot diagrams correspond to the same type knot iff one can be obtained from the other by a finite sequence of Reidemeister moves and plane isotopies.

I I

II II

III Example

II II I Are two diagrams equivalent?

∼? Are two diagrams equivalent?

∼?

∼?

• A knot invariant is a quantity defined for each knot which is the same for equivalent knots.

• A knot or link is tricolorable if we can assign one of three colours to each arc in the knot diagram such that; (1) At least 2 colours are used. (2) At each crossing, either three different colours come together or all the same color comes together. Knot Invariant

• A knot invariant is a quantity defined for each knot which is the same for equivalent knots.

• A knot or link is tricolorable if we can assign one of three colours to each arc in the knot diagram such that; (1) At least 2 colours are used. (2) At each crossing, either three different colours come together or all the same color comes together. Knot Invariant

• A knot invariant is a quantity defined for each knot which is the same for equivalent knots.

• A knot or link is tricolorable if we can assign one of three colours to each arc in the knot diagram such that; (1) At least 2 colours are used. (2) At each crossing, either three different colours come together or all the same color comes together. Knot Invariant

• A knot invariant is a quantity defined for each knot which is the same for equivalent knots.

• A knot or link is tricolorable if we can assign one of three colours to each arc in the knot diagram such that; (1) At least 2 colours are used. (2) At each crossing, either three different colours come together or all the same color comes together. Knot Invariant

• A knot invariant is a quantity defined for each knot which is the same for equivalent knots.

• A knot or link is tricolorable if we can assign one of three colours to each arc in the knot diagram such that; (1) At least 2 colours are used. (2) At each crossing, either three different colours come together or all the same color comes together.

≁ , Knot Invariant

• A knot invariant is a quantity defined for each knot which is the same for equivalent knots.

• A knot or link is tricolorable if we can assign one of three colours to each arc in the knot diagram such that; (1) At least 2 colours are used. (2) At each crossing, either three different colours come together or all the same color comes together.

≁ ≁ , There are many invariants

A knot invariant is a quantity defined for each knot which is the same for equivalent knots.

• Three-Dimensional Invariants - Arc index, Braid index, Bridge index, Crossing number, Determinant, Polygon index, , , , ...

• Polynomial Invariants - , Conway polynomial, HOMFLY polynomial, , Kauffman polynomial, ...

• ··· There are many invariants

A knot invariant is a quantity defined for each knot which is the same for equivalent knots.

• Three-Dimensional Invariants - Arc index, Braid index, Bridge index, Crossing number, Determinant, Polygon index, Tricolorability, Tunnel number, Unknotting number, ...

• Polynomial Invariants - Alexander polynomial, Conway polynomial, HOMFLY polynomial, Jones polynomial, Kauffman polynomial, ...

• ··· There are many invariants

A knot invariant is a quantity defined for each knot which is the same for equivalent knots.

• Three-Dimensional Invariants - Arc index, Braid index, Bridge index, Crossing number, Determinant, Polygon index, Tricolorability, Tunnel number, Unknotting number, ...

• Polynomial Invariants - Alexander polynomial, Conway polynomial, HOMFLY polynomial, Jones polynomial, Kauffman polynomial, ...

• ··· There are many invariants

A knot invariant is a quantity defined for each knot which is the same for equivalent knots.

• Three-Dimensional Invariants - Arc index, Braid index, Bridge index, Crossing number, Determinant, Polygon index, Tricolorability, Tunnel number, Unknotting number, ...

• Polynomial Invariants - Alexander polynomial, Conway polynomial, HOMFLY polynomial, Jones polynomial, Kauffman polynomial, ...

• ··· There are many invariants

A knot invariant is a quantity defined for each knot which is the same for equivalent knots.

• Three-Dimensional Invariants - Arc index, Braid index, Bridge index, Crossing number, Determinant, Polygon index, Tricolorability, Tunnel number, Unknotting number, ...

• Polynomial Invariants - Alexander polynomial, Conway polynomial, HOMFLY polynomial, Jones polynomial, Kauffman polynomial, ...

• ··· Arc presentation and arc index

An arc presentation of a knotor a link L is an embedding of L contained in the union of finitely many half planes, called pages, with a common boundary line, called binding axis, in such a way that each half plane contains a properly embedded single arc.

c d 1 2 3

4 5 b e a

The minimal number of pages among all arc presentations of a link L is called the arc index of L and is denoted by α(L). Arc presentation and arc index

An arc presentation of a knotor a link L is an embedding of L contained in the union of finitely many half planes, called pages, with a common boundary line, called binding axis, in such a way that each half plane contains a properly embedded single arc.

c d 1 2 3

4 5 b e a

The minimal number of pages among all arc presentations of a link L is called the arc index of L and is denoted by α(L). Arc presentation and arc index

An arc presentation of a knotor a link L is an embedding of L contained in the union of finitely many half planes, called pages, with a common boundary line, called binding axis, in such a way that each half plane contains a properly embedded single arc.

c d 1 2 3

4 5 b e a

The minimal number of pages among all arc presentations of a link L is called the arc index of L and is denoted by α(L). Links with arc index up to 5

α(L) 2 3 4 5 L unknot none 2-component , Hopf link trefoil Grid diagram

Cromwell, 1995 Every link admits an arc presentation.

• Every link admits a grid diagram. • A grid diagram gives rise to an arc presentation and vice versa. Grid diagram

Cromwell, 1995 Every link admits an arc presentation.

• Every link admits a grid diagram. • A grid diagram gives rise to an arc presentation and vice versa.

c e b

a d

grid diagram Grid diagram

Cromwell, 1995 Every link admits an arc presentation.

• Every link admits a grid diagram. • A grid diagram gives rise to an arc presentation and vice versa.

c e b

a d

grid diagram Grid diagram

Cromwell, 1995 Every link admits an arc presentation.

• Every link admits a grid diagram. • A grid diagram gives rise to an arc presentation and vice versa.

c e b

a d

grid diagram Grid diagram

Cromwell, 1995 Every link admits an arc presentation.

• Every link admits a grid diagram. • A grid diagram gives rise to an arc presentation and vice versa.

c e b c e

a d b a d grid diagram Grid diagram

Cromwell, 1995 Every link admits an arc presentation.

• Every link admits a grid diagram. • A grid diagram gives rise to an arc presentation and vice versa.

c e b c e

a d b a d grid diagram Grid diagram

Cromwell, 1995 Every link admits an arc presentation.

• Every link admits a grid diagram. • A grid diagram gives rise to an arc presentation and vice versa.

c e b c e

a d b a d grid diagram Grid diagram

Cromwell, 1995 Every link admits an arc presentation.

• Every link admits a grid diagram. • A grid diagram gives rise to an arc presentation and vice versa.

c e b e c a d b a d grid diagram Grid diagram

Cromwell, 1995 Every link admits an arc presentation.

• Every link admits a grid diagram. • A grid diagram gives rise to an arc presentation and vice versa.

c e b e c a d a b d grid diagram Grid diagram

Cromwell, 1995 Every link admits an arc presentation.

• Every link admits a grid diagram. • A grid diagram gives rise to an arc presentation and vice versa.

c e b e c a d b a d grid diagram Grid diagram

Cromwell, 1995 Every link admits an arc presentation.

• Every link admits a grid diagram. • A grid diagram gives rise to an arc presentation and vice versa.

c e b e c a d b a d grid diagram Grid diagram

Cromwell, 1995 Every link admits an arc presentation.

• Every link admits a grid diagram. • A grid diagram gives rise to an arc presentation and vice versa.

c e b e c a d b a d grid diagram Elementary Moves on Grid Diagrams

Dynnikov, 2006 Two grid diagrams of the same link can be obtained from each other by a finite sequence of the following elementary moves. • stabilization and destabilization; • interchanging neighbouring edges if their pairs of endpoints do not interleave; • cyclic permutation of vertical (horizontal) edges. Elementary Moves on Grid Diagrams

Dynnikov, 2006 Two grid diagrams of the same link can be obtained from each other by a finite sequence of the following elementary moves. • stabilization and destabilization; • interchanging neighbouring edges if their pairs of endpoints do not interleave; • cyclic permutation of vertical (horizontal) edges. Elementary Moves on Grid Diagrams

Dynnikov, 2006 Two grid diagrams of the same link can be obtained from each other by a finite sequence of the following elementary moves. • stabilization and destabilization; • interchanging neighbouring edges if their pairs of endpoints do not interleave; • cyclic permutation of vertical (horizontal) edges. Elementary Moves on Grid Diagrams

Dynnikov, 2006 Two grid diagrams of the same link can be obtained from each other by a finite sequence of the following elementary moves. • stabilization and destabilization; • interchanging neighbouring edges if their pairs of endpoints do not interleave; • cyclic permutation of vertical (horizontal) edges. Known Results I

• [Beltrami, 2002] Arc index for prime knots up to 10 crossings are determined. • [Ng, 2006] Arc index for prime knots up to 11 crossings are determined. • [Jin-Lee, 2012] We found a list of 372 knots of 13 crossings and 15 knots of 14 crossings which satisfy α(K) = c(K) − 1. ⋆ [Nutt, 1999] All knots up to arc index 9 are identified. ⋆ [Jin et al., 2006] All prime knots up to arc index 10 are identified. ⋆ [Jin-Park, 2010] All prime knots up to arc index 11 are identified. ⋆ [Jin-Kim] All prime knots up to arc index 12 are identified.(preprint) Known Results I

• [Beltrami, 2002] Arc index for prime knots up to 10 crossings are determined. • [Ng, 2006] Arc index for prime knots up to 11 crossings are determined. • [Jin-Lee, 2012] We found a list of 372 knots of 13 crossings and 15 knots of 14 crossings which satisfy α(K) = c(K) − 1. ⋆ [Nutt, 1999] All knots up to arc index 9 are identified. ⋆ [Jin et al., 2006] All prime knots up to arc index 10 are identified. ⋆ [Jin-Park, 2010] All prime knots up to arc index 11 are identified. ⋆ [Jin-Kim] All prime knots up to arc index 12 are identified.(preprint) Known Results I

• [Beltrami, 2002] Arc index for prime knots up to 10 crossings are determined. • [Ng, 2006] Arc index for prime knots up to 11 crossings are determined. • [Jin-L, 2012] We found a list of 372 knots of 13 crossings and 15 knots of 14 crossings which satisfy α(K) = c(K) − 1. ⋆ [Nutt, 1999] All knots up to arc index 9 are identified. ⋆ [Jin et al., 2006] All prime knots up to arc index 10 are identified. ⋆ [Jin-Park, 2010] All prime knots up to arc index 11 are identified. ⋆ [Jin-Kim] All prime knots up to arc index 12 are identified.(preprint) Known Results I

• [Beltrami, 2002] Arc index for prime knots up to 10 crossings are determined. • [Ng, 2006] Arc index for prime knots up to 11 crossings are determined. • [Jin-Lee, 2012] We found a list of 372 knots of 13 crossings and 15 knots of 14 crossings which satisfy α(K) = c(K) − 1. ⋆ [Nutt, 1999] All knots up to arc index 9 are identified. ⋆ [Jin et al., 2006] All prime knots up to arc index 10 are identified. ⋆ [Jin-Park, 2010] All prime knots up to arc index 11 are identified. ⋆ [Jin-Kim] All prime knots up to arc index 12 are identified.(preprint) Wheel diagram

• Bae-Park described an algorithm which transforms a diagram of a link with n crossings into a wheel diagram with at most n + 2 spokes.

c 1,3 d 1 2,5 2 2,4 3

4 b 5 e a 1,4 3,5

Bae-Park, 2000 If L is a non-split link, then α(L) ≤ c(L) + 2. Wheel diagram

• Bae-Park described an algorithm which transforms a diagram of a link with n crossings into a wheel diagram with at most n + 2 spokes.

c 1,3 d 1 2,5 2 2,4 3

4 b 5 e a 1,4 3,5

Bae-Park, 2000 If L is a non-split link, then α(L) ≤ c(L) + 2. Wheel diagram

• Bae-Park described an algorithm which transforms a diagram of a link with n crossings into a wheel diagram with at most n + 2 spokes.

c 1,3 d 1 2,5 2 2,4 3

4 b 5 e a 1,4 3,5

Bae-Park, 2000 If L is a non-split link, then α(L) ≤ c(L) + 2. A sketch of proof of Bae-Park Theorem

Bae-Park, 2000 If L is a non-split link, then α(L) ≤ c(L) + 2.

Suppose that we have a prime link diagram D with c crossings. There is a sequence of isotopic moves such that the number of crossings is reduced one by one until we get a diagram with c spokes and an extra circle.

Note : The sum of number of regions and spokes is unchanged.

The extra circle can be isotoped to be replaced by two spokes so that we obtain a wheel diagram with c + 2 spokes. A sketch of proof of Bae-Park Theorem

Bae-Park, 2000 If L is a non-split link, then α(L) ≤ c(L) + 2.

Suppose that we have a prime link diagram D with c crossings. There is a sequence of isotopic moves such that the number of crossings is reduced one by one until we get a diagram with c spokes and an extra circle.

Note : The sum of number of regions and spokes is unchanged.

The extra circle can be isotoped to be replaced by two spokes so that we obtain a wheel diagram with c + 2 spokes. A sketch of proof of Bae-Park Theorem

Bae-Park, 2000 If L is a non-split link, then α(L) ≤ c(L) + 2.

Suppose that we have a prime link diagram D with c crossings. There is a sequence of isotopic moves such that the number of crossings is reduced one by one until we get a diagram with c spokes and an extra circle.

Note : The sum of number of regions and spokes is unchanged.

The extra circle can be isotoped to be replaced by two spokes so that we obtain a wheel diagram with c + 2 spokes. A sketch of proof of Bae-Park Theorem

Bae-Park, 2000 If L is a non-split link, then α(L) ≤ c(L) + 2.

Suppose that we have a prime link diagram D with c crossings. There is a sequence of isotopic moves such that the number of crossings is reduced one by one until we get a diagram with c spokes and an extra circle.

2 3 3 2

Note : The sum of number of regions and spokes is unchanged.

The extra circle can be isotoped to be replaced by two spokes so that we obtain a wheel diagram with c + 2 spokes. A sketch of proof of Bae-Park Theorem

Bae-Park, 2000 If L is a non-split link, then α(L) ≤ c(L) + 2.

Suppose that we have a prime link diagram D with c crossings. There is a sequence of isotopic moves such that the number of crossings is reduced one by one until we get a diagram with c spokes and an extra circle.

4 4

2 3 3 2

Note : The sum of number of regions and spokes is unchanged.

The extra circle can be isotoped to be replaced by two spokes so that we obtain a wheel diagram with c + 2 spokes. A sketch of proof of Bae-Park Theorem

Bae-Park, 2000 If L is a non-split link, then α(L) ≤ c(L) + 2.

Suppose that we have a prime link diagram D with c crossings. There is a sequence of isotopic moves such that the number of crossings is reduced one by one until we get a diagram with c spokes and an extra circle.

4 4 4 2 3 2 3 4 3 2 3 2

Note : The sum of number of regions and spokes is unchanged.

The extra circle can be isotoped to be replaced by two spokes so that we obtain a wheel diagram with c + 2 spokes. A sketch of proof of Bae-Park Theorem

Bae-Park, 2000 If L is a non-split link, then α(L) ≤ c(L) + 2.

Suppose that we have a prime link diagram D with c crossings. There is a sequence of isotopic moves such that the number of crossings is reduced one by one until we get a diagram with c spokes and an extra circle.

4 4 4 2 3 2 3 3 2 3 2,4

Note : The sum of number of regions and spokes is unchanged.

The extra circle can be isotoped to be replaced by two spokes so that we obtain a wheel diagram with c + 2 spokes. A sketch of proof of Bae-Park Theorem

Bae-Park, 2000 If L is a non-split link, then α(L) ≤ c(L) + 2.

Suppose that we have a prime link diagram D with c crossings. There is a sequence of isotopic moves such that the number of crossings is reduced one by one until we get a diagram with c spokes and an extra circle.

4 1 4 1 4 2 3 2 3 3 2 3 2,4

Note : The sum of number of regions and spokes is unchanged.

The extra circle can be isotoped to be replaced by two spokes so that we obtain a wheel diagram with c + 2 spokes. A sketch of proof of Bae-Park Theorem

Bae-Park, 2000 If L is a non-split link, then α(L) ≤ c(L) + 2.

Suppose that we have a prime link diagram D with c crossings. There is a sequence of isotopic moves such that the number of crossings is reduced one by one until we get a diagram with c spokes and an extra circle.

4 1 4 1 1 4 4 2 3 2 3 2 3 1 2,4 3 2 3 2,4 3

Note : The sum of number of regions and spokes is unchanged.

The extra circle can be isotoped to be replaced by two spokes so that we obtain a wheel diagram with c + 2 spokes. A sketch of proof of Bae-Park Theorem

Bae-Park, 2000 If L is a non-split link, then α(L) ≤ c(L) + 2.

Suppose that we have a prime link diagram D with c crossings. There is a sequence of isotopic moves such that the number of crossings is reduced one by one until we get a diagram with c spokes and an extra circle.

4 1 1,4 4 1 4 2 3 2 3 2 3 3 2 3 2,4 1,3 2,4

Note : The sum of number of regions and spokes is unchanged.

The extra circle can be isotoped to be replaced by two spokes so that we obtain a wheel diagram with c + 2 spokes. A sketch of proof of Bae-Park Theorem

Bae-Park, 2000 If L is a non-split link, then α(L) ≤ c(L) + 2.

Suppose that we have a prime link diagram D with c crossings. There is a sequence of isotopic moves such that the number of crossings is reduced one by one until we get a diagram with c spokes and an extra circle.

4 1 1,4 4 1 4 2 3 2 3 2 3 3 2 3 2,4 1,3 2,4

Note : The sum of number of regions and spokes is unchanged.

The extra circle can be isotoped to be replaced by two spokes so that we obtain a wheel diagram with c + 2 spokes. A sketch of proof of Bae-Park Theorem

Bae-Park, 2000 If L is a non-split link, then α(L) ≤ c(L) + 2.

Suppose that we have a prime link diagram D with c crossings. There is a sequence of isotopic moves such that the number of crossings is reduced one by one until we get a diagram with c spokes and an extra circle.

4 1 1,4 4 1 4 2 3 2 3 2 3 3 2 3 2,4 1,3 2,4

Note : The sum of number of regions and spokes is unchanged.

The extra circle can be isotoped to be replaced by two spokes so that we obtain a wheel diagram with c + 2 spokes. A sketch of proof of Bae-Park Theorem

Bae-Park, 2000 If L is a non-split link, then α(L) ≤ c(L) + 2.

Suppose that we have a prime link diagram D with c crossings. There is a sequence of isotopic moves such that the number of crossings is reduced one by one until we get a diagram with c spokes and an extra circle.

4 1 1,4 1,4 4 1 4 5 5 3 2 3 2 3 2 3 2 3 2 3 2,4 1,3 2,4 1,3 2,4

Note : The sum of number of regions and spokes is unchanged.

The extra circle can be isotoped to be replaced by two spokes so that we obtain a wheel diagram with c + 2 spokes. A sketch of proof of Bae-Park Theorem

Bae-Park, 2000 If L is a non-split link, then α(L) ≤ c(L) + 2.

Suppose that we have a prime link diagram D with c crossings. There is a sequence of isotopic moves such that the number of crossings is reduced one by one until we get a diagram with c spokes and an extra circle.

4 1 1,4 1,4 4 1 2,5 3,5 4 2 3 2 3 2 3 3 2 3 2,4 1,3 2,4 1,3 2,4

Note : The sum of number of regions and spokes is unchanged.

The extra circle can be isotoped to be replaced by two spokes so that we obtain a wheel diagram with c + 2 spokes. Known Results II

• L : non-split alternating link =⇒ α(L) = c(L) + 2. + ◦ [Morton-Beltrami, 1998] For any link L, α(L) ≥ spreada(FL(a, z)) 2.

◦ [Thistlethwaite, 1988] If L is an alternating link, spreada(FL(a, z)) ≥ c(L). ◦ [Bae-Park, 2000] If L is a non-split link, then α(L) ≤ c(L) + 2.

• = + L : nonalternating ⇒ spreada(FL(a, z)) 2 ≤ α(L) ≤ c(L). + ◦ [M-B] For any link L, α(L) ≥ spreada(FL(a, z)) 2. ◦ [Jin-Park, 2010] A prime link L is nonalternating if and only if

α(L) ≤ c(L).

• [Beltrami, 2002] L : n-semialternating link =⇒ α(L) = c(L) − 2(n − 2). Known Results II

• L : non-split alternating link =⇒ α(L) = c(L) + 2. + ◦ [Morton-Beltrami, 1998] For any link L, α(L) ≥ spreada(FL(a, z)) 2.

◦ [Thistlethwaite, 1988] If L is an alternating link, spreada(FL(a, z)) ≥ c(L). ◦ [Bae-Park, 2000] If L is a non-split link, then α(L) ≤ c(L) + 2.

• = + L : nonalternating ⇒ spreada(FL(a, z)) 2 ≤ α(L) ≤ c(L). + ◦ [M-B] For any link L, α(L) ≥ spreada(FL(a, z)) 2. ◦ [Jin-Park, 2010] A prime link L is nonalternating if and only if

α(L) ≤ c(L).

• [Beltrami, 2002] L : n-semialternating link =⇒ α(L) = c(L) − 2(n − 2). Known Results II

• L : non-split alternating link =⇒ α(L) = c(L) + 2. + ◦ [Morton-Beltrami, 1998] For any link L, α(L) ≥ spreada(FL(a, z)) 2.

◦ [Thistlethwaite, 1988] If L is an alternating link, spreada(FL(a, z)) ≥ c(L). ◦ [Bae-Park, 2000] If L is a non-split link, then α(L) ≤ c(L) + 2.

• = + L : nonalternating ⇒ spreada(FL(a, z)) 2 ≤ α(L) ≤ c(L). + ◦ [M-B] For any link L, α(L) ≥ spreada(FL(a, z)) 2. ◦ [Jin-Park, 2010] A prime link L is nonalternating if and only if

α(L) ≤ c(L).

• [Beltrami, 2002] L : n-semialternating link =⇒ α(L) = c(L) − 2(n − 2). Known Results II

• L : non-split alternating link =⇒ α(L) = c(L) + 2. + ◦ [Morton-Beltrami, 1998] For any link L, α(L) ≥ spreada(FL(a, z)) 2.

◦ [Thistlethwaite, 1988] If L is an alternating link, spreada(FL(a, z)) ≥ c(L). ◦ [Bae-Park, 2000] If L is a non-split link, then α(L) ≤ c(L) + 2.

• = + L : nonalternating ⇒ spreada(FL(a, z)) 2 ≤ α(L) ≤ c(L). + ◦ [M-B] For any link L, α(L) ≥ spreada(FL(a, z)) 2. ◦ [Jin-Park, 2010] A prime link L is nonalternating if and only if

α(L) ≤ c(L).

• [Beltrami, 2002] L : n-semialternating link =⇒ α(L) = c(L) − 2(n − 2). Known Results II

• L : non-split alternating link =⇒ α(L) = c(L) + 2. + ◦ [Morton-Beltrami, 1998] For any link L, α(L) ≥ spreada(FL(a, z)) 2.

◦ [Thistlethwaite, 1988] If L is an alternating link, spreada(FL(a, z)) ≥ c(L). ◦ [Bae-Park, 2000] If L is a non-split link, then α(L) ≤ c(L) + 2.

• = + L : nonalternating ⇒ spreada(FL(a, z)) 2 ≤ α(L) ≤ c(L). + ◦ [M-B] For any link L, α(L) ≥ spreada(FL(a, z)) 2. ◦ [Jin-Park, 2010] A prime link L is nonalternating if and only if

α(L) ≤ c(L).

• [Beltrami, 2002] L : n-semialternating link =⇒ α(L) = c(L) − 2(n − 2). Known Results II

• L : non-split alternating link =⇒ α(L) = c(L) + 2. + ◦ [Morton-Beltrami, 1998] For any link L, α(L) ≥ spreada(FL(a, z)) 2.

◦ [Thistlethwaite, 1988] If L is an alternating link, spreada(FL(a, z)) ≥ c(L). ◦ [Bae-Park, 2000] If L is a non-split link, then α(L) ≤ c(L) + 2.

• = + L : nonalternating ⇒ spreada(FL(a, z)) 2 ≤ α(L) ≤ c(L). + ◦ [M-B] For any link L, α(L) ≥ spreada(FL(a, z)) 2. ◦ [Jin-Park, 2010] A prime link L is nonalternating if and only if

α(L) ≤ c(L).

• [Beltrami, 2002] L : n-semialternating link =⇒ α(L) = c(L) − 2(n − 2). Known Results II

• L : non-split alternating link =⇒ α(L) = c(L) + 2. + ◦ [Morton-Beltrami, 1998] For any link L, α(L) ≥ spreada(FL(a, z)) 2.

◦ [Thistlethwaite, 1988] If L is an alternating link, spreada(FL(a, z)) ≥ c(L). ◦ [Bae-Park, 2000] If L is a non-split link, then α(L) ≤ c(L) + 2.

• = + L : nonalternating ⇒ spreada(FL(a, z)) 2 ≤ α(L) ≤ c(L). + ◦ [M-B] For any link L, α(L) ≥ spreada(FL(a, z)) 2. ◦ [Jin-Park, 2010] A prime link L is nonalternating if and only if

α(L) ≤ c(L).

• [Beltrami, 2002] L : n-semialternating link =⇒ α(L) = c(L) − 2(n − 2). Known Results II

• L : non-split alternating link =⇒ α(L) = c(L) + 2. + ◦ [Morton-Beltrami, 1998] For any link L, α(L) ≥ spreada(FL(a, z)) 2.

◦ [Thistlethwaite, 1988] If L is an alternating link, spreada(FL(a, z)) ≥ c(L). ◦ [Bae-Park, 2000] If L is a non-split link, then α(L) ≤ c(L) + 2.

• = + L : nonalternating ⇒ spreada(FL(a, z)) 2 ≤ α(L) ≤ c(L). + ◦ [M-B] For any link L, α(L) ≥ spreada(FL(a, z)) 2. ◦ [Jin-Park, 2010] A prime link L is nonalternating if and only if

α(L) ≤ c(L).

• [Beltrami, 2002] L : n-semialternating link =⇒ α(L) = c(L) − 2(n − 2). Known Results II

• L : non-split alternating link =⇒ α(L) = c(L) + 2. + ◦ [Morton-Beltrami, 1998] For any link L, α(L) ≥ spreada(FL(a, z)) 2.

◦ [Thistlethwaite, 1988] If L is an alternating link, spreada(FL(a, z)) ≥ c(L). ◦ [Bae-Park, 2000] If L is a non-split link, then α(L) ≤ c(L) + 2.

• = + L : nonalternating ⇒ spreada(FL(a, z)) 2 ≤ α(L) ≤ c(L). + ◦ [M-B] For any link L, α(L) ≥ spreada(FL(a, z)) 2. ◦ [Jin-Park, 2010] A prime link L is nonalternating if and only if

α(L) ≤ c(L).

• [Beltrami, 2002] L : n-semialternating link =⇒ α(L) = c(L) − 2(n − 2). n-semialternating links

A nonsplit n-string tangle T is called strongly alternating if the two closures obtained by joining pairs of nearby endpoints of the strings without creating further crossings are reduced alternating.

⇒ T ⇒

A link L is called n-semialternating if there is a nonalternating diagram D of L which is a tangle sum of two strongly alternating n-tangles. n-semialternating links

A nonsplit n-string tangle T is called strongly alternating if the two closures obtained by joining pairs of nearby endpoints of the strings without creating further crossings are reduced alternating.

T&

⇒ ⇒ T T%

A link L is called n-semialternating if there is a nonalternating diagram D of L which is a tangle sum of two strongly alternating n-tangles. n-semialternating links

A nonsplit n-string tangle T is called strongly alternating if the two closures obtained by joining pairs of nearby endpoints of the strings without creating further crossings are reduced alternating.

T&

⇒ ⇒ T T%

A link L is called n-semialternating if there is a nonalternating diagram D of L which is a tangle sum of two strongly alternating n-tangles. n-semialternating links

A nonsplit n-string tangle T is called strongly alternating if the two closures obtained by joining pairs of nearby endpoints of the strings without creating further crossings are reduced alternating.

T&

⇒ ⇒ T T%

A link L is called n-semialternating if there is a nonalternating diagram D of L which is a tangle sum of two strongly alternating n-tangles. Known Results II

• L : non-split alternating link =⇒ α(L) = c(L) + 2. + ◦ [Morton-Beltrami, 1998] For any link L, α(L) ≥ spreada(FL(a, z)) 2.

◦ [Thistlethwaite, 1988] If L is an alternating link, spreada(FL(a, z)) ≥ c(L). ◦ [Bae-Park, 2000] If L is a non-split link, then α(L) ≤ c(L) + 2. • = + L : nonalternating ⇒ spreada(FL(a, z)) 2 ≤ α(L) ≤ c(L). + ◦ [M-B] For any link L, α(L) ≥ spreada(FL(a, z)) 2. ◦ [Jin-Park, 2010] A prime link L is nonalternating if and only if

α(L) ≤ c(L).

• [Beltrami, 2002] L : n-semialternating link =⇒ α(L) = c(L) − 2(n − 2).

Note : Any alternating diagram with no nugatory crossing and n-semialternating diagramis are adequate. Adequate diagram

A link diagram D is + adequate (resp. − adequate) if (i) D has at least one crossing, and

(ii) in the state s+D (resp. s−D) for each crossing, the two segments e1, e2 (resp. f1, f2) belong to different components of s+D (resp. s−D).

f A s+ D A s- D 2 BB B A e e 1 2 f1

A diagram D is adequate if it is both + adequate and − adequate.

• [Thistlethwaite, 1988] 1. If a link L has an adequate diagram with n crossings, then L cannot be projected with fewer than n crossings. 2. If a link L admits an adequate diagram, the lower bound of

spreada(FL(a, z)) can be calculated. Adequate diagram

A link diagram D is + adequate (resp. − adequate) if (i) D has at least one crossing, and

(ii) in the state s+D (resp. s−D) for each crossing, the two segments e1, e2 (resp. f1, f2) belong to different components of s+D (resp. s−D).

f A s+ D A s- D 2 BB B A e e 1 2 f1

A diagram D is adequate if it is both + adequate and − adequate.

• [Thistlethwaite, 1988] 1. If a link L has an adequate diagram with n crossings, then L cannot be projected with fewer than n crossings. 2. If a link L admits an adequate diagram, the lower bound of

spreada(FL(a, z)) can be calculated. Adequate diagram

A link diagram D is + adequate (resp. − adequate) if (i) D has at least one crossing, and

(ii) in the state s+D (resp. s−D) for each crossing, the two segments e1, e2 (resp. f1, f2) belong to different components of s+D (resp. s−D).

f A s+ D A s- D 2 BB B A e e 1 2 f1

A diagram D is adequate if it is both + adequate and − adequate.

• [Thistlethwaite, 1988] 1. If a link L has an adequate diagram with n crossings, then L cannot be projected with fewer than n crossings. 2. If a link L admits an adequate diagram, the lower bound of

spreada(FL(a, z)) can be calculated. Adequate diagram

A link diagram D is + adequate (resp. − adequate) if (i) D has at least one crossing, and

(ii) in the state s+D (resp. s−D) for each crossing, the two segments e1, e2 (resp. f1, f2) belong to different components of s+D (resp. s−D).

f A s+ D A s- D 2 BB B A e e 1 2 f1

A diagram D is adequate if it is both + adequate and − adequate.

• [Thistlethwaite, 1988] 1. If a link L has an adequate diagram with n crossings, then L cannot be projected with fewer than n crossings. 2. If a link L admits an adequate diagram, the lower bound of

spreada(FL(a, z)) can be calculated. Adequate diagram

A link diagram D is + adequate (resp. − adequate) if (i) D has at least one crossing, and

(ii) in the state s+D (resp. s−D) for each crossing, the two segments e1, e2 (resp. f1, f2) belong to different components of s+D (resp. s−D).

f A s+ D A s- D 2 BB B A e e 1 2 f1

A diagram D is adequate if it is both + adequate and − adequate.

• [Thistlethwaite, 1988] 1. If a link L has an adequate diagram with n crossings, then L cannot be projected with fewer than n crossings. 2. If a link L admits an adequate diagram, the lower bound of

spreada(FL(a, z)) can be calculated. Known Results II

• L : non-split alternating link =⇒ α(L) = c(L) + 2. + ◦ [Morton-Beltrami, 1998] For any link L, α(L) ≥ spreada(FL(a, z)) 2.

◦ [Thistlethwaite, 1988] If L is an alternating link, spreada(FL(a, z)) ≥ c(L). ◦ [Bae-Park, 2000] If L is a non-split link, then α(L) ≤ c(L) + 2. • = + L : nonalternating ⇒ spreada(FL(a, z)) 2 ≤ α(L) ≤ c(L). + ◦ [M-B] For any link L, α(L) ≥ spreada(FL(a, z)) 2. ◦ [Jin-Park, 2010] A prime link L is nonalternating if and only if

α(L) ≤ c(L).

• [Beltrami, 2002] L : n-semialternating link =⇒ α(L) = c(L) − 2(n − 2).

Note : Any alternating diagram with no nugatory crossing and n-semialternating diagramis are adequate. Known Results III

⋆ [L-Jin] Arc index of pretzel knots of type (−p, q, r) (submitted)

⋆ [L] Arc index of Montesinos links of type (−r1, r2, r3)(preprint)

1 ri = a1 + a + ··· + 1 2 a + 1 n−1 an Known Results III

⋆ [L-Jin] Arc index of pretzel knots of type (−p, q, r) (submitted)

⋆ [L] Arc index of Montesinos links of type (−r1, r2, r3)(preprint)

1 ri = a1 + a + ··· + 1 2 a + 1 n−1 an Known Results III

⋆ [L-Jin] Arc index of pretzel knots of type (−p, q, r) (submitted)

⋆ [L] Arc index of Montesinos links of type (−r1, r2, r3) (preprint)

a1

-a R R R 2 1 2 3 a 3 - M ai M ai

-a4 M integer tangles

an 1 ri = a1 + a + ··· + 1 2 a + 1 n−1 an Known Results III

⋆ [L-Jin] Arc index of pretzel knots of type (−p, q, r) (submitted)

⋆ [L] Arc index of Montesinos links of type (−r1, r2, r3) (preprint)

a1

-a R R R 2 1 2 3 a 3 - M ai M ai

-a4 M integer tangles

an 1 ri = a1 + a + ··· + 1 2 a + 1 n−1 an An arc presentation on knot diagrams

Let D be a diagram of a knot or a link L. Suppose that there is a simple closed curve C meeting D in k distinct points which divide D into k arcs α1, α2,...,αk with the following properties:

(1) Each αi has no self-crossings.

(2) If αi crosses over αj at a crossing in RI(resp. RO), then i < j(resp. i > j) and it crosses over αj at any other crossings with αj, respectively (3) For each i, there exists an embedded disk di such that ∂di = C and αi ⊂ di.

(4) di ∩ dj = C, for distinct i and j.

Then the pair (D, C) is called an arc presentation of L with k arcs. C is called the binding circle of the arc presentation. An arc presentation on knot diagrams

Let D be a diagram of a knot or a link L. Suppose that there is a simple closed curve C meeting D in k distinct points which divide D into k arcs α1, α2,...,αk with the following properties:

(1) Each αi has no self-crossings.

(2) If αi crosses over αj at a crossing in 3 RI(resp. RO), then i < j(resp. i > j) and it crosses over αj at any other 2 crossings with αj, respectively 4 (3) For each i, there exists an embedded disk di such that ∂di = C and αi ⊂ di. 1 5 C (4) di ∩ dj = C, for distinct i and j.

Then the pair (D, C) is called an arc presentation of L with k arcs. C is called the binding circle of the arc presentation. An arc presentation on knot diagrams

Let D be a diagram of a knot or a link L. Suppose that there is a simple closed curve C meeting D in k distinct points which divide D into k arcs α1, α2,...,αk with the following properties:

(1) Each αi has no self-crossings.

(2) If αi crosses over αj at a crossing in 3 RI(resp. RO), then i < j(resp. i > j) and it crosses over αj at any other 2 crossings with αj, respectively 4 (3) For each i, there exists an embedded disk di such that ∂di = C and αi ⊂ di. 1 5 C (4) di ∩ dj = C, for distinct i and j.

Then the pair (D, C) is called an arc presentation of L with k arcs. C is called the binding circle of the arc presentation. An arc presentation on knot diagrams

Let D be a diagram of a knot or a link L. Suppose that there is a simple closed curve C meeting D in k distinct points which divide D into k arcs α1, α2,...,αk with the following properties:

(1) Each αi has no self-crossings.

(2) If αi crosses over αj at a crossing in 3 RI(resp. RO), then i < j(resp. i > j) and it crosses over αj at any other 2 crossings with αj, respectively RI 4 (3) For each i, there exists an embedded α1 α2 disk di such that ∂di = C and αi ⊂ di. 1 5 C (4) di ∩ dj = C, for distinct i and j.

Then the pair (D, C) is called an arc presentation of L with k arcs. C is called the binding circle of the arc presentation. An arc presentation on knot diagrams

Let D be a diagram of a knot or a link L. Suppose that there is a simple closed curve C meeting D in k distinct points which divide D into k arcs α1, α2,...,αk with the following properties:

(1) Each αi has no self-crossings. α (2) If αi crosses over αj at a crossing in 4 RO 3 RI(resp. RO), then i < j(resp. i > j) and it crosses over αj at any other α3 2 α crossings with αj, respectively RI 4 5 (3) For each i, there exists an embedded α1 α2 disk di such that ∂di = C and αi ⊂ di. 1 5 C (4) di ∩ dj = C, for distinct i and j.

Then the pair (D, C) is called an arc presentation of L with k arcs. C is called the binding circle of the arc presentation. An arc presentation on knot diagrams

Let D be a diagram of a knot or a link L. Suppose that there is a simple closed curve C meeting D in k distinct points which divide D into k arcs α1, α2,...,αk with the following properties:

(1) Each αi has no self-crossings. α (2) If αi crosses over αj at a crossing in 4 RO 3 RI(resp. RO), then i < j(resp. i > j) and it crosses over αj at any other α3 2 α crossings with αj, respectively RI 4 5 (3) For each i, there exists an embedded α1 α2 disk di such that ∂di = C and αi ⊂ di. 1 5 C (4) di ∩ dj = C, for distinct i and j.

Then the pair (D, C) is called an arc presentation of L with k arcs. C is called the binding circle of the arc presentation. An arc presentation on knot diagrams

Let D be a diagram of a knot or a link L. Suppose that there is a simple closed curve C meeting D in k distinct points which divide D into k arcs α1, α2,...,αk with the following properties:

(1) Each αi has no self-crossings. α (2) If αi crosses over αj at a crossing in 4 RO 3 RI(resp. RO), then i < j(resp. i > j) and it crosses over αj at any other α3 2 α crossings with αj, respectively RI 4 5 (3) For each i, there exists an embedded α1 α2 disk di such that ∂di = C and αi ⊂ di. 1 5 C (4) di ∩ dj = C, for distinct i and j.

Then the pair (D, C) is called an arc presentation of L with k arcs. C is called the binding circle of the arc presentation. An arc presentation on knot diagrams

Let D be a diagram of a knot or a link L. Suppose that there is a simple closed curve C meeting D in k distinct points which divide D into k arcs α1, α2,...,αk with the following properties:

(1) Each αi has no self-crossings. α (2) If αi crosses over αj at a crossing in 4 RO 3 RI(resp. RO), then i < j(resp. i > j) and it crosses over αj at any other α3 2 α crossings with αj, respectively RI 4 5 (3) For each i, there exists an embedded α1 α2 disk di such that ∂di = C and αi ⊂ di. 1 5 C (4) di ∩ dj = C, for distinct i and j.

Then the pair (D, C) is called an arc presentation of L with k arcs. C is called the binding circle of the arc presentation. A representation of the arc presentation

By removing a point P from C away from L, we may identify C \ P with the z-axis and each di \ P with a vertical half plane along the z-axis.

α4 3 1 α3 α1 2 α4 α3 3 2 C 4 α5 4 α2 α1 α2 5 α5 1 p 5

This shows that an arc presentation onto knot diagrams is equivalent to an arc presentation. A representation of the arc presentation

By removing a point P from C away from L, we may identify C \ P with the z-axis and each di \ P with a vertical half plane along the z-axis.

α4 3 1 α3 α1 2 α4 α3 3 2 C 4 α5 4 α2 α1 α2 5 α5 1 p 5

This shows that an arc presentation onto knot diagrams is equivalent to an arc presentation. What are Kanenobu Knots?

p

n =

q n> n< n = 

K(p,q)

Kanenobu, 1986 K(p, q) = K(q, p) and K(p, q)∗ = K(−p, −q). What are Kanenobu Knots?

p

n =

q n> n< n = 

K(p,q)

Kanenobu, 1986 K(p, q) = K(q, p) and K(p, q)∗ = K(−p, −q). K(p, q) = K(q, p)

p p p p

∼ ∼ ∼

q q q q

K(p,q)

p q q ∼ ∼ ∼

q p p

K(q,p) K(p, q) with |p| ≤ q

It is sufficient to consider K(p, q) with |p|≤ q, because ⋆ K(p, q) = K(q, p), ⋆ the arc index of a link and its mirror image are the same, ⋆ K(p, q)∗ = K(−p, −q) = K(−q, −p) K(p, q) with |p| ≤ q

q It is sufficient to consider K(p, q) with |p|≤ q, because ⋆ K(p, q) = K(q, p), ⋆ the arc index of a link and its p mirror image are the same, ⋆ K(p, q)∗ = K(−p, −q) = K(−q, −p) K(p, q) with |p| ≤ q

q It is sufficient to consider K(p, q) with |p|≤ q, because ⋆ K(p, q) = K(q, p), ⋆ the arc index of a link and its p mirror image are the same, ⋆ K(p, q)∗ = K(−p, −q) = K(−q, −p) K(p, q) with |p| ≤ q

q It is sufficient to consider K(p, q) with |p|≤ q, because ⋆ K(p, q) = K(q, p), ⋆ the arc index of a link and its p mirror image are the same, ⋆ K(p, q)∗ = K(−p, −q) = K(−q, −p) K(p, q) with |p| ≤ q

q It is sufficient to consider K(p, q) with |p|≤ q, because ⋆ K(p, q) = K(q, p), ⋆ the arc index of a link and its p mirror image are the same, ⋆ K(p, q)∗ = K(−p, −q) = K(−q, −p) How to determine the arc index of K(p, q)?

Our strategy is ...

• For the lower bound of α(K(p, q)), we compute the spreada(FK(p,q)(a, z)).

Morton-Beltrami, 1998 + For any link K, α(K) ≥ spreada(FK(a, z)) 2.

• For the upper bound of α(K(p, q)), we find arc presentations of K(p, q) with the minimum number of arcs for various values of p and q. How to determine the arc index of K(p, q)?

Our strategy is ...

• For the lower bound of α(K(p, q)), we compute the spreada(FK(p,q)(a, z)).

Morton-Beltrami, 1998 + For any link K, α(K) ≥ spreada(FK(a, z)) 2.

• For the upper bound of α(K(p, q)), we find arc presentations of K(p, q) with the minimum number of arcs for various values of p and q. How to determine the arc index of K(p, q)?

Our strategy is ...

• For the lower bound of α(K(p, q)), we compute the spreada(FK(p,q)(a, z)).

Morton-Beltrami, 1998 + For any link K, α(K) ≥ spreada(FK(a, z)) 2.

• For the upper bound of α(K(p, q)), we find arc presentations of K(p, q) with the minimum number of arcs for various values of p and q. Kauffman polynomial FL(a, z)

The Kauffman polynomial of an oriented knot or link L is defined by

−w(D) FL(a, z) = a ΛD(a, z)

where D is a diagram of L, w(D) the of D and ΛD(a, z) the polynomial determined by the rules (K1), (K2) and (K3).

(K1) ΛO(a, z) = 1 where O is the trivial knot diagram. Λ +Λ = Λ +Λ (K2) D+ (a, z) D− (a, z) z( D0 (a, z) D∞ (a, z)). Λ =Λ = −1Λ (K3) a D⊕ (a, z) D(a, z) a D⊖ (a, z).

D+ D− D0 D∞ D⊕ D D⊖ Example : Λ polynomial of trefoil knot

-1 (K2) Λ + Λ = zΛ + zΛ , (K3) Λ = a ΛΛ, = aΛ

Λ( )= –Λ( )+ zΛ( ) + zΛ( ) D = –Λ( )+zΛ( ) +zΛ( ) +1 > -1 2 -1 = – a + z –Λ +zΛ +zΛ +za -1 [ ( ) ( ) ( ) [ -1>

2 2 2 -1 = za + ( z – 1 ) a +z+ ( z – 2 ) a > -1 w(D) = -3 The a-spread of FL(a, z)

The Laurent degree in the variable a, called the a-spread of the Kauffman = −w(D)Λ polynomial FL(a, z) a D(a, z) is denoted by spreada(FL) and defined by the formula = spreada(FL) max-dega(FL) − min-dega(FL).

= Λ Notice that spreada(FL) spreada( D) for any diagram D of L.

Morton-Beltrami, 1998 Let L be a link. Then + α(L) ≥ spreada(FL) 2. ΛK(p,q)(a, z)

Lemma 1

−1 ΛK(p,q)(a, z) = σp−1σq−1(ΛK(0,0) − 1) + (σpσq+1 − σp−1σq)(ΛK(0,1) − a ) −(p+q) − σpσq(ΛK(−1,1) − 1) + a , where n n  α −β if n > 0  α−β  σn(α, β) =  0 if n = 0  − −  α n−β n  − α−β if n < 0 and α + β = z, αβ = 1. A sketch of proof of Lemma 1

For p ≥ 2, using the skein relations K1, K2, and K3, we have

p−1+q −1 −1 −1 ΛK(p,q) +ΛK(p−2,q) = zΛK(p−1,q) + za (az − 1 + a z ) −(p+q) −(p+q−1) −(p+q−2) = zΛK(p−1,q) + a − za + a .

-1 (K2) Λ + Λ = zΛ + zΛ , (K3) Λ = a ΛΛ, = aΛ

+ = z + z

q q q q A sketch of proof of Lemma 1 (Cont.)

−(p+q) −(p+q−1) −(p+q−2) ΛK(p,q) +ΛK(p−2,q) = zΛK(p−1,q) + a − za + a . p+q Let R(p,q) =ΛK(p,q) − a . Then we have

R(p,q) − zR(p−1,q) + R(p−2,q) = 0. Putting α + β = z and αβ = 1, we have

p−1 R(p,q) − αR(p−1,q) = β(R(p−1,q) − αR(p−2,q)) = β (R(1,q) − αR(0,q)), p−1 R(p,q) − βR(p−1,q) = α(R(p−1,q) − βR(p−2,q)) = α (R(1,q) − βR(0,q)). Therefore, we have

R(p,q) = σpR(1,q) − σp−1R(0,q). In a similar way, we have

R(1,q) = σq+1R(1,0) − σqR(1,−1), R(0,q) = σqR(0,1) − σq−1R(0,0). Therefore, we have

R(p,q) = σp−1σq−1R(0,0) + (σpσq+1 − σp−1σq)R(0,1) − σpσqR(−1,1). Proposition 0 : spreada(FK(p,q)(a, z)).

Morton-Beltrami, 1998 + For any link K, α(K) ≥ spreada(FK(a, z)) 2. Proposition 0 : spreada(FK(p,q)(a, z)).

Morton-Beltrami, 1998 + For any link K, α(K) ≥ spreada(FK(a, z)) 2. Main results

Theorem 1 Let 1 ≤ p ≤ q and pq ≥ 3. Then

α(K(p, q)) = p + q + 6.

Theorem 2 Let p = 0 and q ≥ 3. Then

q + 6 ≤ α(K(0, q)) ≤ q + 7.

Theorem 3 Let p = −1 and q ≥ 3. Then

q + 5 ≤ α(K(−1, q)) ≤ q + 7. Main results

Theorem 1 Let 1 ≤ p ≤ q and pq ≥ 3. Then

α(K(p, q)) = p + q + 6.

Theorem 2 Let p = 0 and q ≥ 3. Then

q + 6 ≤ α(K(0, q)) ≤ q + 7.

Theorem 3 Let p = −1 and q ≥ 3. Then

q + 5 ≤ α(K(−1, q)) ≤ q + 7. Main results

Theorem 1 Let 1 ≤ p ≤ q and pq ≥ 3. Then

α(K(p, q)) = p + q + 6.

Theorem 2 Let p = 0 and q ≥ 3. Then

q + 6 ≤ α(K(0, q)) ≤ q + 7.

Theorem 3 Let p = −1 and q ≥ 3. Then

q + 5 ≤ α(K(−1, q)) ≤ q + 7. Distinguish into five cases Proposition 1. Let 2 ≤ p ≤ q. Then α(K(p, q)) ≤ p + q + 6. Proposition 1. Let 2 ≤ p ≤ q. Then α(K(p, q)) ≤ p + q + 6. Proposition 1. Let 2 ≤ p ≤ q. Then α(K(p, q)) ≤ p + q + 6. Proposition 1. Let 2 ≤ p ≤ q. Then α(K(p, q)) ≤ p + q + 6.

7 6

8 5 10 3 9 4

1 2 Proposition 1. Let 2 ≤ p ≤ q. Then α(K(p, q)) ≤ p + q + 6.

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1 2 10 Proposition 1. Let 2 ≤ p ≤ q. Then α(K(p, q)) ≤ p + q + 6.

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1 2 10 Proposition 1. Let 2 ≤ p ≤ q. Then α(K(p, q)) ≤ p + q + 6.

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1 2 10 Proposition 1. Let 2 ≤ p ≤ q. Then α(K(p, q)) ≤ p + q + 6.

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1 2 10 Proposition 1. Let 2 ≤ p ≤ q. Then α(K(p, q)) ≤ p + q + 6.

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3 7 6 4

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1 2 10 Proposition 1. Let 2 ≤ p ≤ q. Then α(K(p, q)) ≤ p + q + 6.

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2 3 K(2,2) 7 6 4

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8 5 6 10 3 7 9 4 8

9

1 2 10 Proposition 1. Let 2 ≤ p ≤ q. Then α(K(p, q)) ≤ p + q + 6. Proposition 1. Let 2 ≤ p ≤ q. Then α(K(p, q)) ≤ p + q + 6. Proposition 1. Let 2 ≤ p ≤ q. Then α(K(p, q)) ≤ p + q + 6.

10 9 11 12

15 13 8 6 14 7

5 4 3

1 2 Proposition 1. Let 2 ≤ p ≤ q. Then α(K(p, q)) ≤ p + q + 6.

1 10 9 2

11 3 12 4 5 K(4,5) 6 13 8 15 6 7 14 7 8 9 5 10 4 11 12 3 13 1 2 14 15 Proposition 1. Let 2 ≤ p ≤ q. Then α(K(p, q)) ≤ p + q + 6.

q!2 vertical sticks 1 10 9 2

11 3 12 4 5 K(p,q) 6 13 8 15 6 7

8 14 7 p!2 vertical sticks 9 5 10 4 11 12 3 13 1 2 14 15 Proposition 2. Let p = 1 and q ≥ 3. Then α(K(p, q)) = α(K(1, q)) ≤ q + 7. Proposition 2. Let p = 1 and q ≥ 3. Then α(K(p, q)) = α(K(1, q)) ≤ q + 7. Proposition 2. Let p = 1 and q ≥ 3. Then α(K(p, q)) = α(K(1, q)) ≤ q + 7. Proposition 2. Let p = 1 and q ≥ 3. Then α(K(p, q)) = α(K(1, q)) ≤ q + 7. Proposition 2. Let p = 1 and q ≥ 3. Then α(K(p, q)) = α(K(1, q)) ≤ q + 7. Proposition 2. Let p = 1 and q ≥ 3. Then α(K(p, q)) = α(K(1, q)) ≤ q + 7. Proposition 2. Let p = 1 and q ≥ 3. Then α(K(p, q)) = α(K(1, q)) ≤ q + 7. Proposition 2. Let p = 1 and q ≥ 3. Then α(K(p, q)) = α(K(1, q)) ≤ q + 7.

1

9 8 2 K(1,5) 7 3 4 6 5

6

7 4 5 11 2 8 3 9 10 10

12 1 11 12 Proposition 2. Let p = 1 and q ≥ 3. Then α(K(p, q)) = α(K(1, q)) ≤ q + 7.

1

9 8 2 K(1,q) 7 3 4 6 5 q!3 vertical sticks 6

7 4 5 11 2 8 3 9 10 10

12 1 11 12 Theorem 1

Theorem 1 Let 1 ≤ p ≤ q and pq ≥ 3. Then α(K(p, q)) = p + q + 6.

Theorem 1 is proved by Proposition 0 and Proposition 1 and 2. Proposition 3. Let p = 0 and q ≥ 3. Then α(K(p, q)) = α(K(0, q)) ≤ q + 7. Proposition 3. Let p = 0 and q ≥ 3. Then α(K(p, q)) = α(K(0, q)) ≤ q + 7. Proposition 3. Let p = 0 and q ≥ 3. Then α(K(p, q)) = α(K(0, q)) ≤ q + 7. Proposition 3. Let p = 0 and q ≥ 3. Then α(K(p, q)) = α(K(0, q)) ≤ q + 7. Proposition 3. Let p = 0 and q ≥ 3. Then α(K(p, q)) = α(K(0, q)) ≤ q + 7. Proposition 3. Let p = 0 and q ≥ 3. Then α(K(p, q)) = α(K(0, q)) ≤ q + 7. Proposition 3. Let p = 0 and q ≥ 3. Then α(K(p, q)) = α(K(0, q)) ≤ q + 7. Proposition 3. Let p = 0 and q ≥ 3. Then α(K(p, q)) = α(K(0, q)) ≤ q + 7.

1 9 2 K(0,5)

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12 1 11 12 Proposition 3. Let p = 0 and q ≥ 3. Then α(K(p, q)) = α(K(0, q)) ≤ q + 7.

1 9 2 K(0,q)

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7 4 6 5 q!3 vertical sticks 5 6 4 7

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12 1 11 12 Theorem 2

Theorem 2 Let p = 0 and q ≥ 3. Then q + 6 ≤ α(K(0, q)) ≤ q + 7.

Theorem 2 is proved by Proposition 0 and Proposition 3. Proposition 4. Let p = −1 and q ≥ 3. Then α(K(p, q)) = α(K(−1, q)) ≤ q + 7. Proposition 4. Let p = −1 and q ≥ 3. Then α(K(p, q)) = α(K(−1, q)) ≤ q + 7. Proposition 4. Let p = −1 and q ≥ 3. Then α(K(p, q)) = α(K(−1, q)) ≤ q + 7. Proposition 4. Let p = −1 and q ≥ 3. Then α(K(p, q)) = α(K(−1, q)) ≤ q + 7. Proposition 4. Let p = −1 and q ≥ 3. Then α(K(p, q)) = α(K(−1, q)) ≤ q + 7. Proposition 4. Let p = −1 and q ≥ 3. Then α(K(p, q)) = α(K(−1, q)) ≤ q + 7. Proposition 4. Let p = −1 and q ≥ 3. Then α(K(p, q)) = α(K(−1, q)) ≤ q + 7. Proposition 4. Let p = −1 and q ≥ 3. Then α(K(p, q)) = α(K(−1, q)) ≤ q + 7.

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9 8 2 K(-1,5) 7 3 4 6 5

6

4 7 5 11 2 8 3 9 10 10

12 1 11

12 Proposition 4. Let p = −1 and q ≥ 3. Then α(K(p, q)) = α(K(−1, q)) ≤ q + 7.

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12 1 11

12 Theorem 3

Theorem 3 Let p = −1 and q ≥ 3. Then q + 5 ≤ α(K(−1, q)) ≤ q + 7.

Theorem 3 is proved by Proposition 0 and Proposition 4. Proposition 5. Let p ≤ −2 and 2 ≤|p| ≤ q. Then α(K(p, q)) ≤|p| + q + 6. Proposition 5. Let p ≤ −2 and 2 ≤|p| ≤ q. Then α(K(p, q)) ≤|p| + q + 6. Proposition 5. Let p ≤ −2 and 2 ≤|p| ≤ q. Then α(K(p, q)) ≤|p| + q + 6.

10 9 11 12

15 13 8 6 14 7

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1 2 Proposition 5. Let p ≤ −2 and 2 ≤|p| ≤ q. Then α(K(p, q)) ≤|p| + q + 6.

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1 2 15 Proposition 5. Let p ≤ −2 and 2 ≤|p| ≤ q. Then α(K(p, q)) ≤|p| + q + 6.

qT2 vertical sticks 1

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4 12 £p£T2 vertical sticks 3 13 14 1 2 15 Proposition 6. α(K(−2p, 2p)) ≤ 4p + 4 for p ≥ 2.

p!2

p!2

β(K(−2p, 2p)) = 2p + 1 (H.Takioka, On the braid index of Kanenobu knots (submitted)) Proposition 6. α(K(−2p, 2p)) ≤ 4p + 4 for p ≥ 2.

p!2

p!2

β(K(−2p, 2p)) = 2p + 1 (H.Takioka, On the braid index of Kanenobu knots (submitted)) Proposition 6. α(K(−2p, 2p)) ≤ 4p + 4 for p ≥ 2.

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4 3 28 2 27 1 p!2 24 26 23 25 22 20 18 16 19 21 14 15 17 p!2

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β(K(−2p, 2p)) = 2p + 1 (H.Takioka, On the braid index of Kanenobu knots (submitted)) Proposition 6. α(K(−2p, 2p)) ≤ 4p + 4 for p ≥ 2.

1 2 12 9 3 4 11 10 5 8 7 6 6 7 5 8 p!2 9 10 4 11 3 12 ! 28 2 p 2 27 1 13 p!2 24 26 14 23 25 15 22 20 16 18 16 19 21 17 14 17 15 p!2 18 19 13 20 21 22 23 24 25 26 27 28

β(K(−2p, 2p)) = 2p + 1 (H.Takioka, On the braid index of Kanenobu knots (submitted)) The front projection of a Legendrian knot

A knot diagram D represents the front projection of a Legendrian knot if (1) D has no vertical tangencies, (2) the only non-smooth points are generalized cusps, and (3) at each crossing the slope of the overcrossing is smaller than the undercrossing. Thurston-Bennequin number

⋆ A grid diagram gives rise to a Legendrian knot and vice versa.

For a grid diagram G, Thurston-Bennequin number is defined by

tb(G) = ω(G) − c(G) where ω(G) and c(G) are the writhe and the number of southeast corners of G, respectively.

The maximal Thurston-Bennequin number of a knot K, written tb(K), is the maximal tb over all grid diagrams for K. Thurston-Bennequin number

⋆ A grid diagram gives rise to a Legendrian knot and vice versa.

For a grid diagram G, Thurston-Bennequin number is defined by

tb(G) = ω(G) − c(G) where ω(G) and c(G) are the writhe and the number of southeast corners of G, respectively.

The maximal Thurston-Bennequin number of a knot K, written tb(K), is the maximal tb over all grid diagrams for K. Thurston-Bennequin number

⋆ A grid diagram gives rise to a Legendrian knot and vice versa.

+45˚

For a grid diagram G, Thurston-Bennequin number is defined by

tb(G) = ω(G) − c(G) where ω(G) and c(G) are the writhe and the number of southeast corners of G, respectively.

The maximal Thurston-Bennequin number of a knot K, written tb(K), is the maximal tb over all grid diagrams for K. Thurston-Bennequin number

⋆ A grid diagram gives rise to a Legendrian knot and vice versa.

+45˚

For a grid diagram G, Thurston-Bennequin number is defined by

tb(G) = ω(G) − c(G) where ω(G) and c(G) are the writhe and the number of southeast corners of G, respectively.

The maximal Thurston-Bennequin number of a knot K, written tb(K), is the maximal tb over all grid diagrams for K. Thurston-Bennequin number

⋆ A grid diagram gives rise to a Legendrian knot and vice versa.

+45˚

For a grid diagram G, Thurston-Bennequin number is defined by

tb(G) = ω(G) − c(G) where ω(G) and c(G) are the writhe and the number of southeast corners of G, respectively.

The maximal Thurston-Bennequin number of a knot K, written tb(K), is the maximal tb over all grid diagrams for K. Thurston-Bennequin number

⋆ A grid diagram gives rise to a Legendrian knot and vice versa.

+45˚

w#@#06 c#@#6

For a grid diagram G, Thurston-Bennequin number is defined by

tb(G) = ω(G) − c(G) where ω(G) and c(G) are the writhe and the number of southeast corners of G, respectively.

The maximal Thurston-Bennequin number of a knot K, written tb(K), is the maximal tb over all grid diagrams for K. Thurston-Bennequin number

⋆ A grid diagram gives rise to a Legendrian knot and vice versa.

+45˚

w#@#06 c#@#6

For a grid diagram G, Thurston-Bennequin number is defined by

tb(G) = ω(G) − c(G) where ω(G) and c(G) are the writhe and the number of southeast corners of G, respectively.

The maximal Thurston-Bennequin number of a knot K, written tb(K), is the maximal tb over all grid diagrams for K. A relation between α(K) and tb(K)

Matsuda, 2006

−α(K) ≤ tb(K) + tb(K∗), where K∗ is the mirror image of K.

tb(G) = ω(G) − c(G) A relation between α(K) and tb(K)

Matsuda, 2006

−α(K) ≤ tb(K) + tb(K∗), where K∗ is the mirror image of K.

+45˚

tb(G) = ω(G) − c(G) A relation between α(K) and tb(K)

Matsuda, 2006

−α(K) ≤ tb(K) + tb(K∗), where K∗ is the mirror image of K.

+45˚ -45˚

tb(G) = ω(G) − c(G) A relation between α(K) and tb(K)

Matsuda, 2006

−α(K) ≤ tb(K) + tb(K∗), where K∗ is the mirror image of K.

+45˚ -45˚ crossing change

tb(G) = ω(G) − c(G) A relation between α(K) and tb(K)

Matsuda, 2006

−α(K) ≤ tb(K) + tb(K∗), where K∗ is the mirror image of K.

+45˚ -45˚ crossing change

tb(G) = ω(G) − c(G) A relation between α(K) and tb(K)

Matsuda, 2006

−α(K) ≤ tb(K) + tb(K∗), where K∗ is the mirror image of K.

+45˚ -45˚ crossing change

tb(G) = ω(G) − c(G) A relation between α(K) and tb(K)

Matsuda, 2006

−α(K) ≤ tb(K) + tb(K∗), where K∗ is the mirror image of K.

+45˚ -45˚ crossing change

tb(G) = ω(G) − c(G) Question

Question, Ng Does a grid diagram realizing α(K) of a knot K necessarily realize tb(K)? An equivalent statement is that

−α(K) = tb(K) + tb(K∗) for all knots K.

Ng, 2012 The above equality holds for all knots K with 11 or fewer crossings. Question

Question, Ng Does a grid diagram realizing α(K) of a knot K necessarily realize tb(K)? An equivalent statement is that

−α(K) = tb(K) + tb(K∗) for all knots K.

Ng, 2012 The above equality holds for all knots K with 11 or fewer crossings. Thank you.