On Sharp Moves

Saori KANENOBU

Kobe University

Dec. 25, 2014

Saori KANENOBU On Sharp Moves

Contents

I Introduction.

I Background.

I Main theorem.

I Proof of the theorem.

Saori KANENOBU On Sharp Moves

Introduction.

Saori KANENOBU On Sharp Moves

Crossing change [H. Wendt (1937)]

The crossing change is a local move for a knotdiagram:

crossing change

u(K ): the minimum number of crossing changesmaking K into the trivial knot

Saori KANENOBU On Sharp Moves

Theorem. [M.G. Scharlemann (1985)]

u(K ) = 1 =⇒ K : prime.

Example. The trefoil knot is unknotted by a singlecrossing change.

Crossing change

Saori KANENOBU On Sharp Moves

∆ move [H. Murakami, Y. Nakanishi (1989)]

The ∆ move is a local move for a knot diagram:

Δ - move

u∆(K ): the minimum number of ∆ moves makingK into the trivial knot

Saori KANENOBU On Sharp Moves

Example. The trefoil knot is unknotted by a single∆ move.

Δ - move

Question. u∆(K ) = 1 =⇒ K : prime?

Saori KANENOBU On Sharp Moves

# move [H. Murakami (1985)]

The # move is a local move for an oriented knotdiagram:

#

u#(K ): the minimum number of # moves makingK into the trivial knot

Saori KANENOBU On Sharp Moves

Example. The trefoil knot is unknotted by a single# move.

#

Question. u#(K ) = 1 =⇒ K : prime?

Saori KANENOBU On Sharp Moves

Background.

Saori KANENOBU On Sharp Moves

Theorem [H. Murakami-S. Sakai (1993)]

K : trefoil knot, Jn: 2-bridge knot C(4,2n),u#(K ) = 1

=⇒ u#(K ]Jn) = 1.

K J n

2n crossings

Saori KANENOBU On Sharp Moves

Proof.

#

nK # J2n crossings

Saori KANENOBU On Sharp Moves

Local moves generated by a # move

# #

##

(1) (2)

(3) (4)

Saori KANENOBU On Sharp Moves

# gordian distance [H. Murakami (1985)]

K ,K ′: oriented knotd#

G (K , K ′): the minimum number of # movesmaking K into the K ′

Theorem. K ,K ′: oriented knotd#

G (K , K ′)=1 =⇒ Arf (K ) 6= Arf (K ′)

Remark.d#

G (K ,©)=u#(K )

Saori KANENOBU On Sharp Moves

Example.

Arf( )=1

Arf( )=0

Arf( )=1Arf(O)=0

Arf( )=1

(1)

(1)

(1)

31

51

52 41

(4)

Saori KANENOBU On Sharp Moves

Main theorem.

Saori KANENOBU On Sharp Moves

Theorem [H. Murakami-S. Sakai (1993)]

K : trefoil knot, Jn: 2-bridge knot C(4,2n),u#(K ) = 1

=⇒ u#(K ]Jn) = 1.

K J n

2n crossings

Saori KANENOBU On Sharp Moves

Main Theorem

Let n be 3 or an even integer.Then there exist two infinite families of knotsJ1, J2, . . . , Jp, . . . and K1, K2, . . . , Kq, . . .such that:(1) Jp, Kq: n-bridge knots,(2) u#(Jp]Kq) = 1 (p, q = 1, 2, . . . ).

Saori KANENOBU On Sharp Moves

Jp, Kq (n = 2)

Jp: C(-4,2p − 1) Kq: C(4,−2q),

-2q2p-1

J Kp q

Theorem. p = 1 =⇒Theorem[H. Murakami-S. Sakai]

Saori KANENOBU On Sharp Moves

Jp, Kq (n = 3)

J Kp q

-2p

2p+1

q+1

-q q -q-1

Saori KANENOBU On Sharp Moves

Jp, Kq: (n = 4)

2p-1

J Kp q

2q

Saori KANENOBU On Sharp Moves

Proof of the theorem.

Saori KANENOBU On Sharp Moves

Lemma

2n

2n

2n-1#

Saori KANENOBU On Sharp Moves

Proof of main theorem (n = 2)

2p-1

2q -2q

#

J p

K q

Saori KANENOBU On Sharp Moves

Main Theorem

Let n,m be 3 or an even integer.Then there exist two infinite families of knotsJ1, J2, . . . , Jp, . . . and K1, K2, . . . , Kq, . . .such that:(1) Jp, : n-bridge knots,

Kq,: m-bridge knots,(2) u#(Jp]Kq) = 1 (p, q = 1, 2, . . . ).

Saori KANENOBU On Sharp Moves

2-bridge knot # 4-bridge knot

2q

2p-1

#J p

K q

2q

Saori KANENOBU On Sharp Moves