On the minimal coloring number of even-parallels of links Eri - - PowerPoint PPT Presentation

Introduction Theorem Proof of Theorem 2

On the minimal coloring number of even-parallels of links

Eri Matsudo

Nihon University Graduate School of Integrated Basic Sciences

Nihon University, December 20, 2016

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Introduction Theorem Proof of Theorem 2

Z-coloring

Let L be a link, and D a diagram of L. Z-coloring A map C : {arcs of D} → Z is called a Z-coloring on D if it satisfies the condition 2C(a) = C(b) + C(c) at each crossing of D with the over arc a and the under arcs b and c. A Z-coloring which assigns the same color to all the arcs of the diagram is called the trivial Z-coloring. Z-colorable link L is Z-colorable if ∃ a diagram of L with a non-trivial Z-coloring.

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Introduction Theorem Proof of Theorem 2

Let L be a Z-colorable link. Minimal coloring number We difine the minimal coloring number of L, denoted by mincolZ(L), as follows. min{#Im(C) | C : non-trivial Z-coloring on a diagram of L}

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Introduction Theorem Proof of Theorem 2

Let L be a Z-colorable link. Minimal coloring number We difine the minimal coloring number of L, denoted by mincolZ(L), as follows. min{#Im(C) | C : non-trivial Z-coloring on a diagram of L} Theorem [Ichihara-M.] Let L be a non-splittable Z-colorable link. If there exists a simple Z-coloring on a diagram of L, then mincolZ(L) = 4. Theorem [Ichihara-M.] If a non-splittable link L admits a Z-coloring C such that #Im(C) = 5, then mincolZ(L) = 4.

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Introduction Theorem Proof of Theorem 2

Let L be a Z-colorable link. Minimal coloring number We difine the minimal coloring number of L, denoted by mincolZ(L), as follows. min{#Im(C) | C : non-trivial Z-coloring on a diagram of L} Theorem [Ichihara-M.] Let L be a non-splittable Z-colorable link. If there exists a simple Z-coloring on a diagram of L, then mincolZ(L) = 4. Theorem [Ichihara-M.] If a non-splittable link L admits a Z-coloring C such that #Im(C) = 5, then mincolZ(L) = 4. Question For any Z-colorable link L, mincolZ(L) = 4?

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Introduction Theorem Proof of Theorem 2

Parallel of a link For a link L = K1 ∪ · · · ∪ Kc with a diagram D and a set (n1, · · · , nc) of integers ni ≥ 1, we denote by D(n1,··· ,nc) the diagram obtained by taking ni-parallel copies of the i-th component Ki of D on the plane for 1 ≤ i ≤ c. The link L(n1,··· ,nc) represented by D(n1,··· ,nc) is called the (n1, · · · , nc)-parallel of the link L. When L is a knot, we call (n)-parallel L(n) simply an n-parallel, and denote it by Ln.

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Introduction Theorem Proof of Theorem 2

Untwisted 2-parallel A 2-parallel K2 = K1 ∪ K2 of a knot K is called the untwisted 2-parallel where lk(K1, K2) = 0.

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Introduction Theorem Proof of Theorem 2

Theorem 1 The untwisted 2-parallel K2 of a knot K is Z-colorable and mincolZ(K2) = 4. Theorem 2 For any diagram of a c-component link L and any even number n1, · · · , nc at least 4, L(n1,··· ,nc) is Z-colorable and mincolZ(L(n1,··· ,nc)) = 4.

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Introduction Theorem Proof of Theorem 2

Outline of proof of Theorem 2 Let L = K1 ∪ · · · ∪ Kc be a link, and D a diagram of L. We focus on crossings on D(n1,··· ,nc) obtained by taking parallel copies at a crossing of D.

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Introduction Theorem Proof of Theorem 2

We put a circle as fencing the crossings.

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Introduction Theorem Proof of Theorem 2

We put a circle as fencing the crossings. For any parallel arcs (a1, · · · , ak) out of the circle, we fix the colors

• f ak/2 and ak/2+1 are 1 and others are 0.

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Introduction Theorem Proof of Theorem 2

For any arcs inside the circle, we assign colors as follows. In the case nj = 4m + 2(m ∈ N), we assign the colors −1, 0, 1, 2.

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Introduction Theorem Proof of Theorem 2

In the case nj = 4m + 4(m ∈ N), we assign the colors −1, 0, 1, 2, 3.

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Introduction Theorem Proof of Theorem 2

We see that D(n1,··· ,nc) admits a Z-coloring C such that Im(C) = {−1, 0, 1, 2, 3}. Therefore L(n1,··· ,nc) is Z-colorable. Moreover, we eliminate the arcs colored by 3 as follows. It follows mincolZ(L(n1,··· ,nc)) = 4.

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Introduction Theorem Proof of Theorem 2

Thank you for your attention.

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