A COMPUTATION OF SUPERBRIDGE INDEX OF KNOTS
CHOON BAE JEON AND GYO TAEK JIN
Abstract. We show that the list {31 , 41 , 52 , 61 , 62 , 63 , 72 , 73 , 74 , 84 , 87 , 89 } contains all 3-superbridge knots. We also supply the best known estimates of the
superbridge index for all prime knots up to nine crossings.
1. Introduction
A knot is a piecewise smooth simple closed curve embedded in the three dimensional Euclidean space R3 . Two knots are equivalent if there is a piecewise smooth
autohomeomorphism of R3 mapping one knot onto the other. The equivalence class
of a knot K will be called the knot type of K and denoted by [K].
The crookedness of a knot K embedded in R3 with respect to a unit vector ~v is
the number of connected components of the preimage of the set of local maximum
values of the orthogonal projection K → R~v , denoted by b~v (K). Figure 1 illustrates
an example. The superbridge number and the superbridge index of K, denoted by
s(K) and s[K], are defined to be ‘max b~v (K)’ and ‘min max b~v (K)’, respectively,
where the maximum is taken over all unit vectors and the minimum taken over
all equivalent embeddings of K. This invariant was introduced by Kuiper who
computed the superbridge index for all torus knots [8].
R~v
6
K
Figure 1. b~v (K) = 3
The classical invariant bridge index of a knot K, denoted by b[K], is defined to
be the least number of maximal overpass strings in all knot diagrams equivalent
to K [14]. The bridge index can be described by using the crookedness b~v (K).
The bridge number of K, denotd by b(K), is defined to be ‘min b~v (K)’ where the
minimum is taken over all unit vector ~v . The bridge index of K is now obtained
by taking again the minimum over all equivalent embeddings of K, i.e., b[K] =
min min b~v (K). For any non-trivial knots, the superbridge index is greater than
Date: December 11, 2000 (1030).
This work was supported by Brain Korea 21 project.
1
2
C.B. JEON AND G.T. JIN
the bridge index. A knot whose bridge index (resp. superbridge index) is n will be
referred to as an n-bridge knot (resp. n-superbridge knot).
In Section 2, we summarize the authors’ previous work which proved that the
3-superbridge knots are among the 2-bridge knots up to nine crossings other than
the three torus knots of type (2, 5), (2, 7) and (2, 9) which are usually denoted by
51 , 71 and 91 , respectively, following the tables in [1, 13].
In Section 3, we analyze all possible projections of a knot with superbridge number 3 into the plane perpendicular to a straight line meeting the knot in four distinct
points. Finally in Section 4, we tabulate all prime knots up to nine crossings with
the best known estimates of their superbridge index.
pp p p p p p pp
pppqp p ppp
p ppppppppppp
ppp pqp p ppp
ppp p p ppp
p ppppppppppp
(a)
(b)
Figure 2. Two types of projections
p pap p p p p p h
ppp
pp
ppp g
p p p p p p p p p pf
bpp p p p
pp
c ppp
pp p p
d
(1)
e
p pap p p p p p h
ppp
pp
ppp g
p p p p p p p p p pf
bpp p p p
pp
c ppp
pp p p
d
e
(2)
p pap p p p p p
pp¡
¡ phpppp
p
c pp
pp g
pp p
p pp
d ppp p p p ppp f
pp
c ppp
pp p p
(7)
(8)
bpp p p p
e
p pap p p p p p h
ppp
pp
ppp g
p
p
p
ppppppp f
bpp p p p
d
e
p pap p p p p p h
ppp
ppp
ppp g
c pp
pp p p
ppp
p
@
p
p
p
f
p
p
d @p p p
pp
c ppp
pp p p
(3)
(4)
bpp p p p
e
p pap p p p p p h
ppp
pp
ppp g
p p p p p p p p p pf
bpp p p p
d
e
p pap p p p p p
pp @ phpppp
p
c pp
pp p p @p p pppp g
@
ppppppp f
d@
pp
c ppp
pp p p
(9)
(10)
bpp p p p
e
p pap p p p p p h
ppp
pp
ppp g
p
p
p
ppppppp f
bpp p p p
d
p pap p p p p p h
pp
ppp @
@ppppp g
c pp
pp p
p pp
d ppp p p p ppp f
p pap p p p p p h
pp
ppp @¡
@
@ppppp g
c pp
pp ¡
pp
ppp p p @
@
d@
p p p p pp f
p pap p p p p p h
pp
ppp ¡
@
@ppppp g
c pp
pp ¡
p
p pp
d ppp p p p ppp f
pp
c ppp
pp p p
(13)
(14)
(15)
(16)
bpp p p p
e
bpp p p p
e
bpp p p p
e
e
p pap p p p p p h
ppp
pp
ppp g
p p p p p p p p p pf
bpp p p p
d
e
p pap p p p p p h
ppp
pp
ppp g
p p p p p p p p p pf
bpp p p p
pp
c ppp
pp p p
d
(5)
e
p pap p p p p p
ppp @ phpppp
c pp
pp p p @p p pppp g
d ppppppp f
bpp p p p
(6)
e
p pap p p p p p h
ppp
pp
p
ppp g
c pp
p@
p p p p p p p ppp
ppppp f
d@
p pap p p p p p h
pp
pp @
p
@ppppp g
c pp
p@
p p p p p p p pp
ppppp f
d@
(11)
(12)
bpp p p p
e
bpp p p p
e
p pap p p p p p
ppp ¡ phpppp
c pp
pg
pp p¡
ppp
p
p
p
p
p
f
p
p
p
p
d
p
p pap p p p p p
ppp @¡ phpppp
c pp
pg
pp ¡
ppp
@
p
p
p
p
p
p
f
p
p
p
p
d
p
(17)
(18)
bpp p p p
e
bpp p p p
e
Figure 3. Patterns at the quadruple point
@
¡
@
¡
@
¡
@⇔¡
@⇔¡
@
¡
@
¡
@
¡
@
¡
@⇔¡
@⇔¡
@
¡
Figure 4. Relations
⇔ ⇔
A COMPUTATION OF SUPERBRIDGE INDEX
3
2. A rough census of 3-superbridge knots
This section is a summary of the authors’ proof in [5] that 3-superbridge knots
are among the 2-bridge knots up to nine crossings other than the three torus knots
51 , 71 and 91 .
Every non-trivial knot has a quadrisecant, i.e., a straight line that meets the knot
at four distinct points [11, 12]. The projection of a knot with superbridge number
three, into the plane perpendicular to a quadrisecant, is the union of four simple
loops based at a single point with at most six double points. More precisely, the
projection is one of Figure 2 where each rectangle contains a horizontal 2-braid
with at most three crossings. This implies that there are only finitely many knot
types with superbridge index 3. The base point of the four loops in the projection
mentioned is a quadruple point which can be loosened to be one of Figure 3 modulo
the relations in Figure 4. Therefore at most six double points are obtained from
the quadruple point. Replacing each double point with a crossing, we obtain a
knot diagram. The number of crossings in this diagram can be reduced to ten or
less. In the case of ten crossings, the diagram is non-alternating. Since the minimal
crossing diagrams of 2-bridge knots are alternating, we conclude that 3-superbridge
knots are 2-bridge knots with no more than nine crossings. Among the fifty 2-bridge
knots up to nine crossings, we exclude the three torus knots 51 , 71 and 91 which
have superbridge index 4 by [8, Theorem B].
3. A finer census of 3-superbridge knots
Theorem 1. No knots other than 31 , 41 , 52 , 61 , 62 , 63 , 72 , 73 , 74 , 84 , 87 and 89
have superbridge index 3.
In Table A and Table B, we consider the projections obtainable from Figure 2
and Figure 3 modulo the relations of Figure 4. The column headings E, O, EE, EO
and OO indicate that the number of crossings of the 2-braid in the boxes of Figure 2
are even, odd, even-even, even-odd and odd-odd, respectively. The row heading ‘nx’
indicates that the row deals with the combination using Figure 3(n) with the point
labeled with ‘x’ matched with the solid dot in Figure 2. The numbers 2, 3 and 4
indicate that the combination is a link with that many components. The symbols
‘°’, ‘}’ and ‘#’ indicate that the combination is a trivial knot, a possibly trivial
torus knot and a connected sum of two possibly trivial torus knots. The symbol
‘∝’ indicates that the two strings of the 2-braid in one of the boxes of Figure 2(b)
are connected just outside of the box. In this case, the projection reduces to one
of Table A. Since 2-bridge knots are prime and 31 is the only 3-superbridge torus
knot, we only need to examine the remaining combinations which are marked with
bold capital letters. The Tables C–V deal with all possible cases up to reflections.
In each diagram of Tables C–V, each of the four arcs inside the circle is labeled
with ‘x’, ‘y’, ‘z’ or ‘w’ if it has a crossing. In each rectangular box, parallel (resp.
crossed) dotted lines indicate that the 2-braid in it has an even (resp. odd) number
of crossings. The numbers in the column headings [m] and [m, ±n] describe the
2-braids in rectangular boxes in the following way: A zero corresponds to a trivial
braid, a positive number to a right-handed twist with that many crossings and a
4
C.B. JEON AND G.T. JIN
negative number to a left-handed twist with crossings as many as the absolute value
of the number. Each word in the column of ‘Arc levels’ indicates the descending
order of the arcs labeled with the letters in it. For example, the combination of
‘wyxz’ and [2, −2] in Table Q corresponds to the diagram in FIgure 5.
As before, the symbol ‘°’ indicates a trivial knot. The new symbol ‘À’ first
appears in Table D and indicates that there is a 2-braid with three crossings whose
string orientations are not coherent. In this case, any embedding realizing the
corresponding diagram and projection necessarily has superbridge number larger
than three. See the ‘Case 2’ in the proof of [5, Sublemma 7]. As it first appears in
Table E, we slashed on every 51 and 71 because we know that they have superbridge
index 4. Another new symbol ‘c’ first appears in Table P and indicates that a subarc
can be straightened to obtain a torus knot with superbridge index 4. Because this
change does not increase the number of local maxima in any direction, it contradicts
the hypothesis that we started with an embedding of superbridge number three.
1a
1b
2a
2b
2c
2d
2e
2f
2g
2h
3a
3b
3c
3d
3e
3f
3g
3h
4a
4b
4c
e
3
2
2
3
2
°
4
°
2
3
2
2
2
}
3
}
2
2
°
°
3
o
3
}
2
2
2
°
3
°
2
2
2
}
2
2
4
2
2
}
°
°
2
4d
4e
4f
4g
4h
5a
5b
5c
5d
6a
6b
6c
6d
7a
7b
7c
7d
7e
7f
7g
7h
e
°
2
2
°
3
2
°
2
3
2
}
2
2
°
}
2
2
2
}
°
2
o
°
3
3
°
2
2
°
2
2
2
2
2
}
°
2
3
}
3
2
°
}
8a
8b
8c
8d
8e
8f
8g
8h
9a
9b
9c
9d
9e
9f
9g
9h
10a
10b
10c
10d
10e
e
°
2
2
}
3
}
°
2
°
}
3
}
2
2
°
2
°
2
3
2
3
o
°
}
3
2
2
2
°
}
°
2
2
2
3
}
°
}
°
}
2
}
2
10f
10g
10h
11a
11b
11c
11d
11e
11f
11g
11h
12a
12b
12c
12d
13a
13b
13c
13d
13e
13f
e
2
°
2
°
°
2
2
2
°
°
3
2
}
°
}
}
2
}
2
}
}
o
}
°
}
°
°
3
3
3
°
°
2
3
2
°
2
2
}
2
}
2
2
13g
13h
14a
14b
14c
14d
15a
15b
15c
15d
15e
15f
15g
15h
16a
16b
17a
17b
17c
17d
18a
e
}
}
}
2
2
}
}
}
2
2
2
2
}
}
2
C
2
C
}
C
C
Table A. Properties of combinations of Figure 2(a) and Figure 3
ypp p p qpxp p p p p p p pp
ppp
ppppp
pp p p £p£p pppp
z p p p p p p £p w
Figure 5. ‘wyxz’ and [2, −2] in Table Q
o
2
2
2
}
}
2
2
2
}
}
}
}
2
2
}
D
}
D
2
D
D
A COMPUTATION OF SUPERBRIDGE INDEX
5
Throughout the Tables C–V, every knot which first appears is marked with ‘?’.
This proves Theorem 1.
1a
1b
2a
2b
2c
2d
2e
2f
2g
2h
3a
3b
3c
3d
3e
3f
3g
3h
4a
4b
4c
4d
4e
4f
4g
4h
5a
ee
3
2
4
°
2
3
2
3
2
°
3
°
2
2
2
2
2
°
2
2
}
3
∝
∝
3
}
2
eo
2
2
3
°
∝
3
}
2
2
°
2
°
#
2
E
3
3
°
∝
2
2
2
∝
2
4
2
}
oo
#
2
2
°
∝
2
2
2
∝
°
#
°
2
3
F
3
2
°
∝
∝
}
3
2
2
3
}
}
5b
5c
5d
6a
6b
6c
6d
7a
7b
7c
7d
7e
7f
7g
7h
8a
8b
8c
8d
8e
8f
8g
8h
9a
9b
9c
9d
ee
°
2
3
2
°
2
2
2
∝
G
2
G
∝
2
2
3
∝
G
2
#
2
2
}
2
2
#
2
eo
°
2
2
#
°
3
3
#
∝
H
3
2
2
3
}
2
∝
2
2
2
∝
3
2
#
2
2
3
oo
°
}
3
2
°
2
2
2
2
I
4
I
2
2
2
#
2
I
2
3
∝
2
}
2
∝
3
2
9e
9f
9g
9h
10a
10b
10c
10d
10e
10f
10g
10h
11a
11b
11c
11d
11e
11f
11g
11h
12a
12b
12c
12d
13a
13b
13c
ee
G
∝
3
}
3
2
J
2
J
2
3
2
2
∝
}
3
}
∝
2
2
2
}
#
}
G
#
L
eo
H
2
2
2
2
2
K
2
K
∝
2
}
}
∝
}
2
2
2
2
}
E
2
2
}
H
2
M
oo
I
2
#
}
#
∝
2
3
2
∝
#
2
}
2
2
2
2
2
}
2
F
2
3
2
I
3
2
13d
13e
13f
13g
13h
14a
14b
14c
14d
15a
15b
15c
15d
15e
15f
15g
15h
16a
16b
17a
17b
17c
17d
18a
ee
#
G
2
#
2
G
2
2
G
2
2
Q
J
J
Q
2
2
2
L
2
Q
U
Q
U
eo
2
2
N
2
P
H
3
#
2
N
3
R
K
K
2
#
N
T
M
T
2
2
R
2
oo
3
I
O
3
O
I
2
2
I
O
2
S
2
2
S
2
O
2
2
2
S
V
S
V
Table B. Properties of combinations of Figure 2(b) and Figure 3
C
pp pp pp pp pp pp pp
x
ypp p p qp p p p p p p p pp
¡ pppp
pppp @
¡ppp
pp p p @
¡
z p@
p¡
p@
p p p p ppp
w
Arc levels
[0]
[2]
xyzw, xywz, xzwy, xwzy, yxzw, yxwz, yzwx, ywzx,
zxyw, zyxw, zwxy, zwyx, wxyz, wyxz, wzxy, wzyx
°
°
xzyw, ywxz, wxzy
°
? 41
? 31
? 52
yzxw, zxwy, wyzx
°
31
zywx
31
°
xwyz
Table C
6
C.B. JEON AND G.T. JIN
D
pp pp p p p pp pp
x
ypp p p qp p p p p p p p pp
¡ pppp
pppp @
¡ppp
¡
pp p p @
z p@
p¡
p@
p p p p ppp
w
Arc levels
[1]
[3]
xyzw, xywz, xzwy, xwzy, yxzw, yxwz, yzwx, ywzx,
zxyw, zyxw, zwxy, zwyx, wxyz, wyxz, wzxy, wzyx
°
°
xzyw, ywxz, wxzy
31
À
xwyz
41
À
yzxw, zxwy, zywx, wyzx
°
À
Table D
E
Arc levels
pp pp pp pp pp pp pp
pp pp p p p pp ppp
[0,1]
[0,3]
[2,1]
[2,−1]
[2,3]
[2,−3]
xy
°
31
°
31
°
6 51
yx
°
31
41
°
? 62
52
[1,1]
[1,−1]
[1,±3]
[3,±3]
xy
°
°
À
À
yx
31
°
À
À
p qpxp p p p p p
yp³
³Pp pp
pp P
ppppp
pp p p
ppp
pp
p
p
p
pp p p p ppp
Table E
F
Arc levels
pp pp p p p pp pp
pp pp p p p pp pp
p qpxp p p p p p
y³
³Pp pp
pp p P
ppppp
pp p p
pppppp
ppp
pp
p
p
p pp
Table F
G
pp pp pp pp pp pp pp
Arc levels
pp pp pp pp pp pp pp
pxp p p p p p p
ypp p p qP
p
pppp ¡Pppppp
pp¡
pp
zp p p p p p p p p p p p p
[0,0]
[0,2]
[2,2]
[2,−2]
xyz, yxz
°
°
°
52
xzy
°
31
°
°
yzx
°
41
62
6 51
zxy, zyx
°
°
41
31
Table G
A COMPUTATION OF SUPERBRIDGE INDEX
H
Arc levels
pp pp pp pp pp pp pp
pp pp p p p pp ppp
p qpxp p p p p p
ypp p PPp pp
pppppB ¡ ppppp
p¡ p
zp pBp p p p p p p p p p p
7
[0,1]
[0,3]
[2,1]
[2,−1]
[2,±3]
xyz, yxz
°
°
°
41
À
xzy
°
41
°
°
À
yzx
31
52
52
°
À
°
°
31
°
À
[1,1]
[1,−1]
[1,3]
[1,−3]
[3,±3]
xyz, yxz
°
31
°
6 51
À
xzy, zxy, zyx
°
°
31
31
À
yzx
41
°
62
52
À
zxy, zyx
Table H
I
Arc levels
pp pp p p p pp pp
pp pp p p p pp pp
pxp p p p p p p
ypp p p qP
ppp ¡Pppppp
pppB¡ ppp
zp pBp p p p p p p p p p p
Table I
J
pp pp pp pp pp pp pp
Arc levels
pp pp pp pp pp pp pp
x
ypp p p qp p p p p p p p pp
ppppp ¡ ppppp
p¡ p
zp p p p p p p p p p p p p
[0,0]
[0,2]
[2,2]
[2,−2]
xyz, zyx
°
°
31
41
xzy
°
41
°
°
yxz
°
°
°
? 61
yzx
°
31
52
61
zxy
°
°
52
°
Table J
K
pp pp pp pp pp pp pp
Arc levels
pp pp p p p pp ppp
x
ypp p p qp p p p p p p p pp
ppppp ¡ ppppp
p¡ p
zp p p p p p p p p p p p p
[0,1]
[0,3]
[2,1]
[2,−1]
[2,±3]
xyz, zyx
°
°
°
31
À
xzy
31
52
°
°
À
yxz
°
°
31
52
À
yzx
°
41
31
6 51
À
°
°
41
°
À
zxy
Table K
8
C.B. JEON AND G.T. JIN
L
pp pp pp pp pp pp pp
Arc levels
pp pp pp pp pp pp pp
x
ypp p p qp p p p p p p p pp
¡ pppp
pppppB @
@£pp
p¡
p
p
p
z Bp p p p p p £p p p p
w
[0,0]
[0,2]
[2,2]
[2,−2]
xyzw, xywz, xwyz, yxzw, yxwz,
zywx, zwxy, zwyx, wzxy, wzyx
°
°
31
41
xzyw, xzwy, xwzy
°
41
°
°
yzxw
31
52
? 74
°
yzwx, ywzx, wyzx
°
31
52
61
ywxz, wxyz, wyxz
°
°
°
61
zxyw, zxwy, zyxw
°
°
52
°
wxzy
31
°
°
°
Table L
M
pp pp pp pp pp pp pp
Arc levels
pp pp p p p pp ppp
x
ypp p p qp p p p p p p p pp
¡ pppp
pppppB @
@£pp
p¡
p
p
z Bp p p p p p p £p p p p
w
[0,1]
[0,3]
[2,1]
[2,−1]
[2,±3]
xyzw, xywz, xwyz, yxzw,
yxwz, zywx, zwxy, zwyx,
wzxy, wzyx
°
°
°
31
À
xzyw, xzwy, xwzy
31
52
°
°
À
yzxw
41
61
62
31
À
yzwx, ywzx, wyzx
°
41
31
6 51
À
ywxz, wxyz, wyxz
°
°
31
52
À
zxyw, zxwy, zyxw
°
°
41
°
À
wxzy
°
31
°
°
À
[0,1]
[0,3]
[2,1]
[2,−1]
[2,3]
xyzw, yxzw, yzxw
°
°
°
52
41
? 73
xwzy
°
52
°
°
31
31
yzwx
41
62
61
°
? 84
? 72
wxzy, wzxy, wzyx
31
6 51
°
°
31
31
Table M
N
pp pp pp pp pp pp pp
Arc levels
pp pp p p p pp ppp
pxp p p p p p p
ypp p p qP
Ppppp
pppppB
p
p
ppp
zp pBp p p p p p p p p p p
w
[2,−3]
Reducible to “xy” in Table E : xywz, xzyw, xzwy, xwyz, yxwz, zxyw, zxwy, zyxw
Reducible to “yx” in Table E : ywxz, ywzx, zywx, zwxy, zwyx, wxyz, wyxz, wyzx
Table N
A COMPUTATION OF SUPERBRIDGE INDEX
O
Arc levels
pp pp p p p pp pp
pp pp p p p pp pp
p qpxp p p p p p
ypp p PPp pp
ppp
pppppB
pp
pp p p
p
p
p
z Bp p p p p p p p
w
9
[1,1]
[1,−1]
[1,±3]
[3,±3]
xyzw, yxzw, yzxw
°
41
À
À
xwzy
°
31
À
À
yzwx
52
°
À
À
wxzy,wzxy,wzyx
°
°
À
À
Reducible to “xy” in Table F : xywz, xzyw, xzwy, xwyz, yxwz, zxyw, zxwy, zyxw
Reducible to “yx” in Table F : ywxz, ywzx, zywx, zwxy, zwyx, wxyz, wyxz, wyzx
Table O
P
pp pp pp pp pp pp pp
Arc levels
pp pp p p p pp ppp
pxp p p p
y³
p p p q ³p p p ppp
ppppp
p
£ppppp
ppp
p
p
p
p
z p p p p p p £p
w
[0,1]
[0,3]
[2,1]
[2,−1]
[2,3]
[2,−3]
xyzw, xzyw, xzwy
31
6 51
°
°
c
31
yxzw
31
6 51
62
41
c
°
ywzx, wyzx, wzyx
°
°
°
52
41
c
wzxy
°
°
31
6 51
°
c
Reducible to “xy” in Table E : xywz, xwyz, xwzy, zxyw, zxwy, zwxy, wxyz, wxzy
Reducible to “yx” in Table E : yxwz, yzxw, yzwx, ywxz, zyxw, zywx, zwyx, wyxz
Table P
Q
pp pp pp pp pp pp pp
Arc levels
pp pp pp pp pp pp pp
x
ypp p p qp p p p p p p p pp
pppp ¡ pppp
pp¡ £pp
zp p p p p p p p p £p p p p
w
Reducible
Reducible
Reducible
Reducible
to
to
to
to
“xyz”
“xzy”
“yzx”
“zxy”
[0,0]
[0,2]
[2,2]
[2,−2]
xyzw, xywz
°
°
41
31
xzyw
31
52
°
°
zxyw
°
°
61
°
zwyx, wzyx
°
°
°
52
wyxz
°
°
31
72
wyzx
31
°
31
73
in
in
in
in
Table
Table
Table
Table
G
G
G
G
:
:
:
:
xwyz, yxzw, yxwz, ywxz, wxyz
xzwy, xwzy, wxzy
yzxw, yzwx, ywzx
zxwy, zyxw, zywx, zwxy, wzxy
Table Q
10
C.B. JEON AND G.T. JIN
R
Arc levels
pp pp pp pp pp pp pp
ypp p
pp pp p p p pp pp
p qpxp p p p p p
p pp
ppppp ¡ £ ppppp
p¡ p
zp p p p p p p p p £p p p p
w
[0,1]
[0,3]
[2,1]
[2,−1]
[2,±3]
xyzw, xywz
°
°
31
°
À
xzyw
41
61
°
°
À
zxyw
°
°
52
31
À
zwyx, wzyx
°
°
°
41
À
wyxz
°
°
41
61
À
°
31
°
61
À
wyzx
Reducible
Reducible
Reducible
Reducible
to
to
to
to
“xyz”
“xzy”
“yzx”
“zxy”
in
in
in
in
Table
Table
Table
Table
H
H
H
H
:
:
:
:
xwyz, yxzw, yxwz, ywxz, wxyz
xzwy, xwzy, wxzy
yzxw, yzwx, ywzx
zxwy, zyxw, zywx, zwxy, wzxy
Table R
S
pp pp p p p pp pp
Arc levels
pp pp p p p pp pp
x
ypp p p qp p p p p p p p pp
ppppp ¡ £ ppppp
p¡ p
zp p p p p p p p p £p p p p
w
[1,1]
[1,−1]
[1,3]
[1,−3]
[3,±3]
xyzw, xywz
°
°
31
31
À
xzyw, zxyw
31
°
6 51
°
À
zwyx, wzyx
°
31
°
6 51
À
wyxz
31
6 51
°
6 71
À
wyzx
°
52
41
73
À
Reducible to “xyz” in Table I : xwyz, yxzw, yxwz, ywxz, wxyz
Reducible to “xzy” in Table I : xzwy, xwzy, wxzy, zxwy, zyxw, zywx, zwxy, wzxy
Reducible to “yzx” in Table I : yzxw, yzwx, ywzx
Table S
T
Arc levels
pp pp pp pp pp pp pp
ypp p
pp pp p p p pp pp
p qpxp p p p p p
ppppp
pp p p
z pppppp
p pp
ppp
pp
pp p pp
w
[0,1]
[0,3]
[2,1]
[2,−1]
[2,3]
[2,−3]
xyzw, xzyw
°
°
°
°
c
31
xywz, zxyw
°
31
°
41
c
62
xzwy
41
62
°
°
c
31
yxzw
°
°
31
? 63
c
? 87
ywzx
°
52
31
62
c
? 89
zwyx, wyxz
°
31
52
31
c
63
wyzx, wzyx
31
6 51
6 51
31
c
°
wzxy
31
6 51
52
°
c
41
} or ° : xwyz, xwzy, yxwz, yzxw, yzwx, ywxz, zxwy, zyxw, zywx, zwxy, wxyz, wxzy
Table T
A COMPUTATION OF SUPERBRIDGE INDEX
U
pp pp pp pp pp pp pp
Arc levels
pp pp pp pp pp pp pp
x
ypp p p qp p p p p p p p pp
ppppp
pp p p
z ppppppp
ppp
pp
p
p pp
w
11
[0,0]
[0,2]
[2,2]
[2,−2]
xyzw
°
°
c
°
xywz
°
31
c
41
xzyw
31
°
°
31
xzwy
°
41
°
31
yxzw
°
°
c
63
ywzx
°
52
c
62
zxyw
°
°
°
62
zwyx
°
°
52
63
wyxz
°
31
c
31
wyzx
31
6 51
c
31
wzxy
°
31
52
41
wzyx
°
31
6 51
°
} or ° : xwyz, xwzy, yxwz, yzxw, yzwx, ywxz, zxwy, zyxw, zywx, zwxy, wxyz, wxzy
Table U
V
pp pp p p p pp pp
Arc levels
pp pp p p p pp pp
x
ypp p p qp p p p p p p p pp
ppp
ppp
ppp
pp
p
p
p
z pp p p p p p p p pp
w
[1,1]
[1,−1]
[1,3]
[1,−3]
[3,3]
[3,−3]
xyzw
°
°
c
31
c
°
xywz
°
°
c
31
c
63
xzyw
°
31
°
6 51
c
31
xzwy
°
°
31
31
c
62
yxzw
°
41
c
62
c
89
ywzx
52
31
c
63
c
87
zxyw
°
31
°
6 51
c
87
zwyx
°
41
52
62
c
89
wyxz
52
°
c
41
c
62
wyzx
6 51
31
c
°
c
31
wzxy
°
°
31
31
c
63
wzyx
31
°
6 51
°
c
°
} or ° : xwyz, xwzy, yxwz, yzxw, yzwx, ywxz, zxwy, zyxw, zywx, zwxy, wxyz, wxzy
Table V
12
C.B. JEON AND G.T. JIN
4. Primes knots up to nine crossings
There are a number of upper bounds for the superbridge index.
UB1
UB2
UB3
UB4
Twice the braid index [8].
One half of the polygon index [6].
The harmonic index [15].
Five times the bridge index minus three [4].
The p[K] in the fourth column of Table W is the polygon index of K which is
the minimal number of straight edges needed to form a polygonal knot equivalent
to K. The values and the estimates of polygon index in the table were collected
from various articles [2, 3, 6, 10]. Roughly speaking, any local extrema can occur
only at vertices and hence no more than one half of the vertices can attain local
maxima. Except 91 , the value or the upper bound of s[K] is obtained from UB2.
For 91 , UB1 applies. Since the harmonic index is known for only a handful of
K
31
41
51
52
61
62
63
71
72
73
74
75
76
77
81
82
83
84
85
86
87
88
89
810
811
812
813
814
b[K]
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
2
2
2
2
3
2
2
2
2
s[K]
3
3
4
3–4
3–4
3–4
3–4
4
3–4
3–4
3–4
4
4
4
4–5
4–5
4–6
3–5
4–6
4–6
3–6
4–5
3–6
4–6
4–5
4–6
4–5
4–5
p[K]
6
7
8
8
8
8
8
9
9
9
9
9
9
9
9–10
9–11
9–12
9–10
9–12
9–12
9–12
9–11
9–12
9–12
9–10
9–12
9–11
9–11
K
815
816
817
818
819
820
821
91
92
93
94
95
96
97
98
99
910
911
912
913
914
915
916
917
918
919
920
921
b[K]
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
2
2
2
2
2
s[K]
4–6
4
4
4
4
4
4
4
4–7
4–6
4–7
4–6
4–6
4–6
4–6
4–6
4–6
4–6
4–6
4–6
4–7
4–5
4–7
4–7
4–6
4–6
4–6
4–7
p[K]
9–12
9
9
8–9
8
8
9
9–13
9–14
9–12
9–14
9–13
9–13
9–12
9–13
9–13
9–13
9–13
9–12
9–13
9–14
9–11
9–14
9–14
9–13
9–13
9–13
9–14
K
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
Table W. Prime knots up to 9 crossings
b[K]
3
2
3
3
2
2
3
3
3
2
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
s[K]
4–7
4–7
4–6
4–7
4–6
4–6
4–6
4–7
4–6
4–6
4–6
4–6
4–6
4–6
4–7
4–7
4–7
4–6
4
4
4
4–5
4–5
4–5
4
4–6
4–6
4–5
p[K]
9–14
9–14
9–12
9–15
9–12
9–12
9–12
9–15
9–13
9–13
9–12
9–12
9–12
9–13
9–14
9–14
9–15
9–13
9
9
9
9–10
9–10
9–10
9
9–12
9–12
9–11
A COMPUTATION OF SUPERBRIDGE INDEX
13
knots, UB3 shows no significant role in this work. For 2-bridge knots and 3-bridge
knots, UB4 gives 7 and 12, respectively, hence no better bounds than UB2.
Finally, we would like to conjecture that the population of 3-superbridge knots
is two.
References
[1] G. Burde and H. Zieschang, Knots, de Gruyter Studies in Mathematics, vol. 5, Walter de
Gruyter, Berlin, New York, 1985.
[2] J. Calvo, Geometric knot theory: the classification of spatial polygons with a small number
of edges, Ph.D. thesis, University of California Santa Barbara, 1998.
[3] J. Calvo and K.C. Millett, Minimal edge piecewise linear knots, preprint, 1998.
[4] E. Furstenberg, J. Li and J. Schneider, Stick knots, Chaos, Solitons & Fractals 9(1998)
561–568.
[5] C.B. Jeon and G.T. Jin, There are only finitely many 3-superbridge knots, J. Knot Theory
Ramifications (Special issue of Knots in Hellas 1998), to appear.
[6] G.T. Jin, Polygon indices and superbridge indices of torus knots and links, J. Knot Theory
Ramifications 6(1997) 281–289.
[7] G.T. Jin, Superbridge index of composite knots, J. Knot Theory Ramifications 9(2000) 669–
682.
[8] N. Kuiper, A new knot invariant, Math. Ann. 278(1987) 193–209.
[9] G. Kuperberg, Quadrisecants of knots and links, J. Knot Theory Ramifications 3(1994)
41–50.
[10] M. Meissen, Lower and upper bounds on edge numbers and crossing numbers of knots, Ph.D.
thesis, University of Iowa, 1997.
[11] H.R. Morton and D.M.Q. Mond, Closed curves with no quadrisecants, Topology 21(1982)
235–243.
[12] E. Pannwitz, Eine elementargeometrisch Eigenshaft von Verschlingungen und Knoten,
Math. Ann. 108(1933) 629–672.
[13] D. Rolfsen, Knots and Links, Mathematics Lecture Series 7, Publish or Perish, 1976.
[14] H. Schubert, Über eine numerische Knoteninvariente, Math. Zeit. 61(1954) 245–288.
[15] Aaron Trautwein, Harmonic knots, Ph.D. Thesis, University of Iowa, 1994.
Department of Mathematics, KAIST, Taejon 305-701 Korea
E-mail address: [email protected], [email protected]