仮想結び目のチェッカーボード彩色に ついて
広島大学大学院理学研究科 今別府 孝規
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Def
A virtual link diagram is a link diagram which may have virtual crossings.
positive negative virtual
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Def
A virtual link is the equivalence class of virtual link diagram under the generalized Reidemeister moves.
Generalized Reidmeister moves
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Def
Σ : a compact oriented surface.
D : a virtual link diagram, which is a deformation retract of Σ.
A pair (Σ, D) is called an abstract link diagram (ALD).
Def
φ :{virtual link diagram}−→{ALD}
We call φ(D) an ALD associated with D.
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ex
Virtual link diagram ALD
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Def
A checkerboard coloring of ALD is a coloring of regions by black and white, such that adjacent regions have different colorings.
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Def
D : a virtual link diagram.
If an ALD associated with D is checkerboard colorable, then D is checkerboard colorable.
Fact
If D is classical link diagram, then D is checkerboard colorable.
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Def
L : a virtual link.
L is checkerboard colorable
⇐⇒ L has a checkerboard colorable virtual link diagram.
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A table of virtual knots with up to four real crossing [N.Kamada]
1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18
19 20 21 22 23 24 25 26 27
29 30 31 32 33 34 35 36
28
38 39 40 41 42 43 44
37 45
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46 47 48 49 50 51 52 53 54
55 56 57 58 59 60 61 62 63
64 65 66 67 68 69 70 71 72
73 74 75 76 77 78 79 80 81
83 84 85 86 87 88 89 90
82
91 92 93 94 95 96 97 98 99
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Result 1
The following 13 virtual knots are checkerboard colorable.
1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18
19 20 21 22 23 24 25 26 27
29 30 31 32 33 34 35 36
28
38 39 40 41 42 43 44
37 45
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Result 1
The following 13 virtual knots are checkerboard colorable.
1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18
19 20 21 22 23 24 25 26 27
29 30 31 32 33 34 35 36
28
38 39 40 41 42 43 44
37 45
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46 47 48 49 50 51 52 53 54
55 56 57 58 59 60 61 62 63
64 65 66 67 68 69 70 71 72
73 74 75 76 77 78 79 80 81
83 84 85 86 87 88 89 90
82
91 92 93 94 95 96 97 98 99
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46 47 48 49 50 51 52 53 54
55 56 57 58 59 60 61 62 63
64 65 66 67 68 69 70 71 72
73 74 75 76 77 78 79 80 81
83 84 85 86 87 88 89 90
82
91 92 93 94 95 96 97 98 99
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2 3 4
38 39 40
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41 42 43
44 45 46
47
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Result 2
The following 83 virtual knots are not checkerboard colorable.
1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18
19 20 21 22 23 24 25 26 27
29 30 31 32 33 34 35 36
28
38 39 40 41 42 43 44
37 45
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Result 2
The following 83 virtual knots are not checkerboard colorable.
1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18
19 20 21 22 23 24 25 26 27
29 30 31 32 33 34 35 36
28
38 39 40 41 42 43 44
37 45
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46 47 48 49 50 51 52 53 54
55 56 57 58 59 60 61 62 63
64 65 66 67 68 69 70 71 72
73 74 75 76 77 78 79 80 81
83 84 85 86 87 88 89 90
82
91 92 93 94 95 96 97 98 99
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46 47 48 49 50 51 52 53 54
55 56 57 58 59 60 61 62 63
64 65 66 67 68 69 70 71 72
73 74 75 76 77 78 79 80 81
83 84 85 86 87 88 89 90
82
91 92 93 94 95 96 97 98 99
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Fact[Kauffman ’99]f-polynomial f: {virtual link}−→ Z[A, A−1]
Thm[N.Kamada ’02]
L: a virtual link with n components.
EXP(f(L)): the set of integers appearing as exponents of f(L).
ex) f(L) = 3A6 + A4 − 2A−2 EXP(f(L)) = {6, 4, −2}
If L is checkerboard colorable, then EXP(f(L)) ⊂ 4Z if n is odd, and EXP(f(L)) ⊂ 4Z + 2 if n is even.
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ex
No.1
f(L) = A−4 + A−6 − A−10 EXP(f(L)) = {−4, −6, −10}
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Def
c : a classical crossing of an oriented virtual link diagram D.
Dc:a virtual link daigram of the component which have c.
Smoothing c
c
Dc0 : a virtual link diagram of the smoothed Dc.
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If Dc0 is a two-component virtual link diagram, we choose an ordering (k1,k2) for the components of Dc0.
V : the set of virtual crossings between k1 and k2.
k1 k1
k2 k2
ind(v) = 1 ind(v) = −1
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c
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k2 k1
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Def
If Dc0 is a two-component virtual link diagram, then the virtual intersection index i(c) is given by
i(c) = | X
v∈V
ind(v)|.
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k2 k1
i(c1) = 2
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Def
If Dc0 is a one-component virtual link diagram, then i(c) is given by
i(c) = 0.
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c
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i(c2) = 0
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Thm[Y.H.Im-K.Lee-S.Y.Lee ’10]
D : a virtual link diagram.
A polynomial Q(D) is defined by Q(D) = X
c
sign(c)(ti(c) − 1) ∈ Z[t].
Q(D) is an invariant for virtual links.
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c
1i(c1) = 2, sign(c1)(ti(c1) − 1) = t2 − 1
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c
2i(c2) = 0, sign(c2)(ti(c2) − 1) = 0
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Q(D) = t2 + 2t − 3
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Thm[Y.H.Im-K.Lee-S.Y.Lee ’10]
L : a virtual link.
If L is checkerboard colorable, then Q(L) ∈ Z[t2].
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Q(D) = t2 + 2t − 3 ∈/ Z[t2]
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Result 2
The following 83 virtual knots are not checkerboard colorable.
1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18
19 20 21 22 23 24 25 26 27
29 30 31 32 33 34 35 36
28
38 39 40 41 42 43 44
37 45
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Result 2
The following 83 virtual knots are not checkerboard colorable.
1 5 6 8 9
10 11 12 13 14 15 16 17 18
19 20 21 22 23 24 25 26 27
29 30 31 32 33 34 35 36
37
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Result 2
The following 83 virtual knots are not checkerboard colorable.
1 5 6 8 9
10 11 12 13 14 15 16 17 18
19 20 21 22 23 24 25 26 27
29 30 31 32 33 34 35 36
37
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46 47 48 49 50 51 52 53 54
55 56 57 58 59 60 61 62 63
64 65 66 67 68 69 70 71 72
73 74 75 76 77 78 79 80 81
83 84 85 86 87 88 89 90
82
91 92 93 94 95 96 97 98 99
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48 49 50 51 52 53 54
55 56 57 58 59 60 62 63
64 65 66 67 68 69 70 71 72
73 74 75 76 77 78 79 80 81
83 84 85 86 87 88 89 90
82
91 92 93 94 95 96 97 98 99
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48 49 50 51 52 53 54
55 56 57 58 59 60 62 63
64 65 66 67 68 69 70 71 72
73 74 75 76 77 78 79 80 81
83 84 85 86 87 88 89 90
82
91 92 93 94 95 96 97 98 99
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Result 3
For the following 3 virtual knots, checkerboard colorability is unknown.
7 28 96
f = 1 f = 1 f = 1 − A−4
Q = 0 Q = 0 Q = 2(t2 − 1)
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Thank you
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