toshifumi tanaka gifu universityjin/meetings/2020/ttanaka... · 2020. 2. 20. · toshifumi tanaka...
TRANSCRIPT
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Symmetric unions and essential tori
10:55–11:35, Monday 17 February, 2020
Workshop “Knots and Spatial Graphs 2020” KAIST, Daejeon Korea
Toshifumi Tanaka
Gifu University
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Def. A link is a disjoint union of embedded circles
in S 3.
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Def.If K1 and K2 are knots, the connected sum
K1#K2 is defined as follows:
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The connected sum
We have two non-parallel essential tori in the complement.
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The connected sum
We have two non-parallel essential tori in the complement.
T
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We have an essential torus in the complement of the pattern link J ⋃ K1.
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Satellite knotDef.Let Vh be a solid torus which is the complement of the unknot J in S 3.
Let Kh be a knot in Vh such that Kh is a geometrically essential.
We define the order of the pair (Vh, Kh) as the geometric intersection number
of Kh to any meridian disk of Vh.
Let Vc be a regular neighborhood of a non-trivial knot Kc in S 3.
We call a knot K is a satellite knot if K is the image Φ(Kh) for a homeo.
Φ :Vh Vc , the order of (Vh, Kh) is not zero and Kh is not the core of Vh.
J ⋃ Kh : Pattern link
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Symmetric Union
A symmetric union D ⋃D *(n1, ∙∙∙ , nk ) (ℤ ∋ ni ≠ ∞)
is defined by the following diagram:
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Symmetric Union
Tangles
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Property
Fact.
(1) D ⋃D *( n ) ( S. Kinoshita-H. Terasaka (OMJ 1957))
(2) D ⋃D *(n1, ∙∙∙ , nk ) (C. Lamm (OJM 2000))
Fact (Lamm).(1) Every symmetric union is a ribbon knot.
(2)△(D ⋃D *(n1, ∙∙∙ , nk )) =△(D ⋃D *(n’1, ∙∙∙ , n’k ))
if ni ≡ ni’ (mod 2) for all i.
(△:Alexander polynomial. )
(3) det(D ⋃D *(n1, ∙∙∙ , nk )) = det(D )2 .
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Minimal twisting number
Def.
We call the number of non-zero elements in {n1, ∙∙∙ , nk }
the twisting number for D ⋃D *(n1, ∙∙∙ , nk ).
The minimal twisting number of a symmetric union K is the
smallest number of the twisting numbers of all symmetric
union presentations to K denoted by tw(K).
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tw (K) = 1.
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Prop. (T. JKTR (2019)).
Let K1 and K2 be prime symmetric union with
tw(K1) = tw(K2) = 1.
Then
tw(K1 # K2 ) = 2 iff K1 is not the mirror image of K2.
Fact
K1, K2 ∈ {61, 88, 820, 946, 103, 1022, 1035, 10137, 10140, 10153}
⇒ tw (K1 # K2 ) = 2.
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Prop. (T. JKMS (2015)).
tw (1099) = tw (10123)= 2.
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Th. (T. JKTR (2019)).
There are infinitely many symmetric unions with
minimal twisting number two.
m > 1
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Ribbon knot
Def. A ribbon knot is a knot that bounds a self-
intersecting disk with only ribbon singularities.
Ribbon singularity
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Open problem
Every symmetric union is a ribbon knot.
Problem (Lamm OJM (2000)).Is every ribbon knot, symmetric union?
(1) Every ribbon knot with crossing number≦10 is
a symmetric union.
(2) Every two-bridge ribbon knot is a symmetric union.
C. Lamm, The search for non-symmetric ribbon knots, arXiv:1710.06909, 2017.
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Potential counterexampleC. Lamm, The search for non-symmetric ribbon knots, arXiv:1710.06909, 2017.
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Th. (T).
Let K be a satellite symmetric union with minimal
twisting number one.
If the order of the pattern of K is an odd number ≧ 3,
then the complement of K has two disjoint non-parallel
essential tori which are symmetric with respect to a
plane.
In particular, the pattern link complement contains an
essential torus.
Conj. The condition for minimum twisting number
is unnecessary.
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Example
K
K is a symmetric union with minimal twisting number ≤ 1.
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Example
K
K is a prime knot ⇒ K has minimal twisting number one.
W. B. R. Lickorish, Prime knots and tangles,
Trans. Amer. Math. Soc. 267 (1981)
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Example
Pattern The order is 3.
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Example
Pattern link
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Example
Let Kc be a non-trivial ribbon knot as a companion knot
⇒ the satellite knot K is ribbon knot.
The order of
this pattern is 3.
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Example
Theorem ⇒ The satellite knot K is not a symmetric union
with minimal twisting number one.
The computer
program HIKMOT,
which is integrated
into SnapPy by M.
Culler, N. Dunfield
and J. Weeks,
shows that the
pattern link J ⋃ Kh
is hyperbolic and
in particular it does
not contain an
essential torus.
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Reference1. S. Kinoshita and H.Terasaka, On unions of knots, Osaka J. Math. Vol. 9
(1957), 131-153.
2. C. Lamm, Symmetric unions and ribbon knots, Osaka J. Math., Vol. 37
(2000), 537-550.
3. T. Tanaka, The Jones polynomial of knots with symmetric union
presentations, J. Korean Math. Soc. 52 (2015), no. 2, 389-402.
4. C. Lamm, The search for non-symmetric ribbon knots, arXiv:1710.06909,
2017.
5. T. Tanaka, On composite knots with symmetric union presentations, J. Knot Theory
Ramifications 28 (2019), no. 10.
Thank you for your attention