Tohoku Knot Seminar (-2006 in Late Autumn-), 15.Nov.’06
Crossing numbers and
bounds on the boundary slope sets for Montesinos knots
市原一裕 Kazuhiro Ichihara
大阪産業大学 Osaka Sangyo Univ.
Notation
K : a knot in S3
E(K) : the exterior of K,
i.e., E(K) = S3 − open tubular nbhd of K Definition (Essential surface)
An embedded surface F in E(K) is called essential if F is incompressible and ∂-incompressible.
Remark:
Surfaces are not assumed to be orientable.
§1. Diameter of Boundary slope set
Definition (Slope)
A slope on ∂E(K) := the isotopy class of non-trivial simple closed curves on ∂E(K).
Let F be an essential surface in E(K).
Definition (Boundary slope)
The ∂-slope of F := the slope determined by
Definition (Boundary slope set)
For a knot K, the set of ∂-slopes is called
the boundary slope set. We denote it by BK.
Facts:
[Hatcher, ’82]
∂-slopes are only finitely many.
[Culler-Shalen, ’84]
There are at least two ∂-slopes.
⇒ BK is
a non-empty, finite set.
We fix;
the standard meridian-longitude system on ∂E(K).
Then slopes are parametrized by;
rational numbers Q and 10.
i.e.
{
slope on ∂E(K)
} 1:1
←→ Q ∪ {10}
For a knot K,
non-meridional elements in B gives
Example (figure-eight knot)
+4 −4
∂-slopes are {−4, 0, 4}
(“0” indicates the ∂-slope of Seifert surfaces).
Definition (Boundary slope diameter) For a knot K, Diam(K) is defined as
the difference between the greatest & the least in BK − {10}.
Facts.
(Culler-Shalen) Diam(K) ≥ 2 if 1
0 6∈ BK
(Ishikawa-Mattman-Shimokawa)
Diam(K) > ||β||
q||µ||
§2. Diam(K) vs Crossing number Let Cr(K) denote
the minimal crossing number of a knot K in S3. Question
Is there a relationship
between Diam(K) and Cr(K)?
This is motivated by
Ishikawa-Shimokawa’s observations.
Torus knot
For a non-trivial torus knot Tp,q,
BTp,q = {0, pq} ; Cr(Tp,q) = pq − p.
We have;
2 Cr(K) ≥ Diam(K) ≥ Cr(K) + 3.
(Observed by T. Mattman)
Alternating knot
Fact (Aumann ’56, Delman-Roberts ’99) For any prime alternating knots,
two checkerboard surfaces are both essential.
Observation:
Difference of such ∂-slopes = 2 Cr(K).
Proposition
Diam(K) ≥ 2 Cr(K) holds
for a prime alternating knot K.
Two-bridge knot and link
Theorem [Mattman-Maybrun-Robinson]
Diam(K) = 2 Cr(K) holds for a two-bridge knot K.
Theorem [Hoste-Shanahan]
Diam∆(L) = 2 Cr(L) holds for a two-bridge link L.
Montesinos knot
Theorem [I.-Mizushima]
Diam(K) ≤ 2 Cr(K) holds for a Montesinos knot K.
Corollary
In particular, Diam(K) = 2 Cr(K) holds for an alternating Montesinos knot K.
Conjecture
In general, Diam(K) ≤ 2 Cr(K) holds.
In particular;
2 Cr(K) = Diam(K) holds if K is alternating.
Question
Is there a lower bound on Diam(K)
§3. Result
Theorem [I.-Mizushima, (NEW!) ]
Diam(K) ≥ 2 Cr(K) − 6 holds
for a non-trivial Montesinos knot K.
Problem: Is it SHARP?
Definition (Rational tangle)
A tangle consists of two strings in a 3-ball, and is called rational tangle if the strings comes from arcs with slope p/q on a four-punctured sphere.
Example
h23i We assume that
each slope is non-integral ( just for normalization).
Montesinos knot :
A knot K in the 3-sphere S3 obtained by
connecting a number of rational tangles in line:
Remark
We assume that;
the number of tangles is at least 3.
From this, we eliminate
two-bridge knots from our arguments.
§4. Outline of Proof
For a Montesinos knot K = M(T1, T2, . . . , Tn).
Embedded surface in E(K) m
Edgepath system in D Reference:
A. Hatcher and U. Oertel,
Boundary slopes for Montesinos knots, Topology 28 (1989), no. 4, 453-480.
The diagram D is
certain 1-dim. cell complex embedded in the plane.
·· · ·
- 6
x y
O
In fact, D is regarded as
certain subset of the space of the measured laminations
on the 4-punctured sphere.
·· · ·· ·
- 6
x y
O
Example: edgepath system in D
x 1
y
x 1
y
x 1
y
Twist of edgepath system
Let Γ be an edgepath system.
The twist τ(Γ) is defined as (roughly) 2∑
(signed lengths of edges in Γ).
Then the ∂-slope R of the surface associated to Γ is calculated as
R = τ(Γ) − τ(ΓSeifert)
where ΓSeifert denotes the edgepath system associated to a Seifert surface.
Let
Γinc: monotonically increasing edgepath system Γdec: monotonically decreasing edgepath system for a Montesinos knot K.
Proposition [I.-Mizushima]
|τ(Γinc)| + |τ(Γdec)| = 2 Cr(K).
Assume:
Both Γinc and Γdec correspond to some essential surfaces in E(K).
In this case, these surfaces actually give
the greatest and the least ∂-slopes Rmax, Rmin. Thus,
Diam(K) = |Rmax − Rmin| = |τ (Γinc) − τ(Γdec)|
= |τ(Γinc)| + |τ (Γdec)|
= 2 Cr(K)
Unfortunately surfaces coming from edgepath systems are often inessential.
If the monotonic edgepath systems correspond to inessential surfaces,
consider “nearly” monotonic edgepath systems which correspond to essential surfaces,
and analyze them in detail...
Case 1 K satisfies Λdec(0) ≥ 0.
Case 2 K satisfies Λdec(0) = −1.
Case 2-1 K satisfies ]{i|ri = −1} = 0.
Case 2-2 K satisfies ]{i|ri = −1} = 1.
Case 2-2-1 K satisfies ]{i|ri ≤ −3} = 0.
Case 2-2-2 K satisfies ]{i|ri ≤ −3} = 1.
Case 2-2-2-1 K satisfies ]{i|ri = −2} = 1 (i.e., N = 3).
Case a K satisfies −1/3 ≤ T1 < 0.
Case b K satisfies −1/2 ≤ T1 < −1/3.
Case b-a K satisfies r3 = −3 or −4.
Case b-b K satisfies r3 ≤ −5.
Case 2-2-2-2 K satisfies ]{i|ri = −2} ≥ 2 (i.e., N ≥ 4).
Case 2-2-3 K satisfies ]{i|ri ≤ −3} ≥ 2.
Case 2-3 K satisfies ]{i|ri = −1} ≥ 2.
Case 2-3-1 Λdec of K satisfies (*) in Proposition 2.9 [H-O].
Case 2-3-2 Λdec of K does not satisfy (*) in Proposition 2.9 in [H-O].
Case 3 K satisfies Λdec(0) ≤ −2.
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November 27 – November 29, 2006 Osaka Sangyo University,
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