Introduction Known results Main theorem
On the minimal coloring number of the minimal diagram of torus links
Eri Matsudo
Nihon University
Graduate School of Integrated Basic Sciences
Joint work with
K. Ichihara (Nihon Univ.) & K. Ishikawa(RIMS, Kyoto Univ.) Waseda University, December 24, 2018
Introduction Known results Main theorem
Z-coloring
LetL be a link, andD a diagram ofL.
Z-coloring
A mapγ :{arcs ofD} →Zis called a Z-coloring onDif it satisfies the condition2γ(a) =γ(b) +γ(c)at each crossing of D with the over arcaand the under arcsb andc.
AZ-coloring which assigns the same color to all the arcs of the diagram is called atrivialZ-coloring.
Lis Z-colorableif∃ a diagram ofL with a non-trivialZ-coloring.
Introduction Known results Main theorem
LetL be a Z-colorable link.
Minimal coloring number [1] For a diagramDof L,
mincolZ(D) := min{#Im(γ)|γ :non-tri. Z-coloring onD}
[2] mincolZ(L) := min{mincolZ(D)|D:a diagram ofL}
Introduction Known results Main theorem
SimpleZ-coloring
γ : aZ-coloring on a diagram Dof a non-trivial Z-colorable link L If∃d∈Ns.t. at each crossings in D, the differences between the colors of the over arcs and the under arcs aredor 0, then we callγ asimpleZ-coloring.
Theorem 1 [Ichihara-M., JKTR, 2017]
LetL be a non-splittableZ-colorable link. If there exists a simple Z-coloring on a diagram of L, then mincolZ(L) = 4.
Introduction Known results Main theorem
SimpleZ-coloring
γ : aZ-coloring on a diagram Dof a non-trivial Z-colorable link L If∃d∈Ns.t. at each crossings in D, the differences between the colors of the over arcs and the under arcs aredor 0, then we callγ asimpleZ-coloring.
Theorem 1 [Ichihara-M., JKTR, 2017]
LetL be a non-splittableZ-colorable link. If there exists a simple Z-coloring on a diagram of L, then mincolZ(L) = 4.
Introduction Known results Main theorem
Theorem 2 [M., to apper JKTR, Zhang-Jin-Deng]
AnyZ-colorable link has a diagram admitting a simpleZ-coloring.
Colorally
L: aZ-colorable link mincolZ(L) =
2 (L : splittable) 4 (L : non-splittable)
Introduction Known results Main theorem
Theorem 2 [M., to apper JKTR, Zhang-Jin-Deng]
AnyZ-colorable link has a diagram admitting a simpleZ-coloring.
Colorally
L: aZ-colorable link mincolZ(L) =
2 (L : splittable) 4 (L : non-splittable)
Introduction Known results Main theorem
One of key moves of the proof
→The obtained diagrams are often complicated. Problem
mincolZ(Dm) =? for aminimal diagram Dm of a Z-colorable link.
Introduction Known results Main theorem
One of key moves of the proof
→The obtained diagrams are often complicated. Problem
mincolZ(Dm) =? for aminimal diagram Dm of a Z-colorable link.
Introduction Known results Main theorem
One of key moves of the proof
→The obtained diagrams are often complicated.
Problem
mincolZ(Dm) =? for aminimal diagram Dm of a Z-colorable link.
Introduction Known results Main theorem
One of key moves of the proof
→The obtained diagrams are often complicated.
Problem
mincolZ(Dm) =? for aminimal diagram Dm of a Z-colorable link.
Introduction Known results Main theorem
Theorem 3 [Ichihara-M., Proc.Inst.Nat.Sci., Nihon Univ., 2018]
[1] For an even integern≥2, the pretzel linkP(n,−n,· · ·, n,−n) with at least4 strands has a minimal diagramDm s.t.
mincolZ(Dm) =n+ 2.
[2] For an integern≥2, the pretzel linkP(−n, n+ 1, n(n+ 1)) has a minimal diagramDm s.t. mincolZ(Dm) =n2+n+ 3.
[3] For even integern >2 and non-zero integerp, the torus link T(pn, n) has a minimal diagramDm s.t. mincolZ(Dm) = 4.
Introduction Known results Main theorem
Theorem 4 [Ichihara-Ishikawa-M., In progress]
Letp, q andr be non-zero integers such that |p| ≥q ≥1and r≥2. If pr orqr are even, the torus linkT(pr, qr)has a minimal diagramDm s.t.
mincolZ(Dm) =
4 (r : even)
00500 (r : odd) Remark
A torus linkT(pr, qr)isZ-colorable if and only ifpror qrare even.
Introduction Known results Main theorem
[Proof of Theorem 4 (In the caser:even)]
LetDbe the following minimal diagram of T(pr, qr).
Introduction Known results Main theorem
In the following, we will find aZ-coloringγ onDby assigning colors on the arcs ofD.
We devide such arcs intoq subfamilies x1,· · ·,xq.
Introduction Known results Main theorem
We first find a localZ-coloringγ. In the case r is even, we start with settingγ(xi) = (γ(xi,1), γ(xi,2),· · · , γ(xi,r))
= (1,0,· · · ,0,1)for anyi.
Introduction Known results Main theorem
We can extendγ on the arcs in the regions (1)and(q+ 1).
Introduction Known results Main theorem
We can extendγ on the arcs in the regions (2),(3),· · · ,(q).
Introduction Known results Main theorem
Now,γ can be extended on all the arcs in the region depicted as follows.
SinceDis composed of pcopies of the local diagram, it concludes thatD admits aZ-coloring with only four colors0,1,2and 3.
Introduction Known results Main theorem
Now,γ can be extended on all the arcs in the region depicted as follows.
Introduction Known results Main theorem
