only amphicheirals in. Reidemeister's table are the seven identified by Tait in 1884. Diagrams of the 165 prime. 10-crossing knot types are appended. (Murasugi.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 45, Number 2, August
1974
ON THE CLASSIFICATION OF KNOTS KENNETH A. PERKO, JR. ABSTRACT. Linking numbers between branch curves of irregular covering spaces of knots are used to extend the classification of knots through ten crossings and to show that the only amphicheirals in Reidemeister's table are the seven identified by Tait in 1884. Diagrams of the 165 prime 10-crossing knot types are appended. (Murasugi and the author have proven them prime; Conway claims proof that the tables are complete.) Including the trivial type, there are precisely 250 prime knots with ten or fewer crossings.
This
knots
paper
tabled
completes
by Tait
the classification
[9] and Little
for the six remaining
knots
(by knot type)
[4] and solves
in Reidemeister's
of the
10-crossing
the amphicheirality
table
problem
[6, pp. 70—72].
The method of proof is that set forth in [6, Chapter III, §15]-Iinking numbers
between
branch
[3, §5 and Reference The coverings
curves
considered
regular
p-gon)
stricted
to representations
sheeted
covering
the curve
dihedral
between
x with the curve linking
number
Received
group
sending
between
sum of those
ber between
are those
spaces.
1 See also
of index and
on four letters.
curve
1. Similarly,
by the editors
of branching v
1 and the sum of those
of index
of index
of a
and
v.
of index number
of index
is
2 and the
is the linking
num-
2 in a p-
in such
a
2 which have linking 2 which
number have
1.
July 3, 1973 and, in revised
AMS(MOS)subject classifications
of a
are re-
index
(p odd)
1 and the sum of all curves
y with the curve
The latter
into transpositions
v Ax, y) is the linking
the sum of all curves of index
out of representations
2p (the group of symmetries
meridians
the branch
of index
covering
arising
group of order
or the symmetric
number
algebraic
covering
13].
knot group onto the dihedral
the linking
of irregular
form, September
12, 1973.
(1970). Primary 55A25, 55A10.
Key words and phrases. Knots with ten crossings, amphicheirality, linking numbers between branch curves of irregular covering spaces of knots. 1 These linking numbers may be computed geometrically as discussed in the author's paper On covering spaces of knots, to appear in Glasnik Matematicki 9 (29) (1974). Calculations with respect to particular examples are available on request. Copyright © 1974, American Mathematical Society
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
262
263
ON THE CLASSIFICATION OF KNOTS Since
the index
a branched numbers
(taken
topological
induced
invariants
there
the set of such
to a simultaneous
are known
viously
onto a given reversal
units
nonamphicheiral
numbers
and a representational
table
of 2-bridged
include
knots
group)
table,
63, 8,,
means,
theorems
prove
non-
orientation with
8g, 812, 817 and
[l,
The linking
six have
including
pre-
the
[7, p. 170], signatures
dihedral
of Bürde
has been extended
919, 923 and 9,,.)
also
of
linking are
is nonsymmetric
4.,
by various
in 3-sheeted
invariant
which
(for a particular
[8] and all but the following
[6, p. 31], Schubert's
[5, p. 400], linking
of Si D k, such
of sign.
to be amphicheiral
been proven
Minkowski
autohomeomorphisms
of k in S numbers
Of the 84 knots in Reidemeister's 8,„
under
is more than one representation)
of the complement
where
and all representations
respect
is preserved
by autohomeomorphisms
as a set where
amphicheirality of S
of branching
covering
covering
p. 120].
(Note
by Murasugi
numbers tabled
spaces that
[3, p. 200] Schubert's
[unpublished]
to
below prove that
all six are nonamphicheiral.
820
v9=±2,
930
iy3 =±8,
924
v9=±2,
933
v6l(6,-6)=
Of the 166 10-crossing been previously
table)
by the rational
fractions
told, to the parameters coverings)
type,
[2].
together
Table
(both
either
associated
1 sets
shown
in Figure
1, the pair
tected
duplication
in Little's
(8 times),
949
f, = ±16/5
(6 times).
from each
by their
with 2-bridged
10
knots
examples,
which
10
newly
turns
all but 31 have
and from the knots
their grouped
2-sheeted
1
we are cyclic
by polynomial
30 of them.
out to be a previously
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
in
or, in 3 cases,
(and corresponding,
distinguish
table.
Figure
tables,
polynomials
which constitute
forth these
numbers
other
Alexander
of the lens spaces
with linking
iy = ±6/7
knots in the Tait-Little
distinguished
Reidemeister's
±4,
941
unde-
As
264
K. A. PERKO, JR.
Table 1
10,
±4
10,
+4
10an
5,
±4
7s
±4
10,
±8
10,
±8
±22/3 ±22/3
±4
±22/3 10, 10,
±6
10,
±2
10,
±4
102
±6
10,
±6/5
±4
I0„,
±6
±4/5 ±4/5 ±4/5 ±4/5 ±4/5 +4/5 104
1046
±10
±2 10,
±4
±6/5
820
0
±■1
±4 ±4
±4 ±4
8,6
±12
±4
±6
10, 10,
10.,v
±6
±4
10«
U4/5
10, 928
±4
929
±10
103VJ
10,
±12
10,
±24/5 ±24/5 ±24/5 ±24/5 ±24/5 +56/5
±10/3 114/3 ±14/3
±2
10,
ho
+4
'O4V,,,
+12
±2
10, 93a
±14
10.„ u69
±4
»o« 0
10,
102
10, 1077 940
«
1042
±2
10,
±4
±52/11
"29(2.-2)
10,
0 su
«2/11
6VI
«2/11
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
10,
±10
265
ON THE CLASSIFICATION OF KNOTS
PRIME KNOTS WITH TEN CROSSINGS Taken from the tables
of P. G. Tait
and C. N. Little
[Trans. Roy. Soc. Edinburgh 32 (1885) and 39 (1900)] (The knots
duplication
in Little's
with two bridges
table
has been omitted and
are grouped
at the beginning.)
1
3
11
14
15
16
20
21
22
24
37
41
43
48
49
51
52
58
60
61
62
65
66
67
68
71
74
76
77
78
79
80
81
101
102
104
105
106
107
108
117
119
120
121
122
2
4
5
6
7
8
9
10
12
13
17
18
19
23
25
26
27
28
29
30
31
32
33
34
35
36
38
39
40
42
44
45
46
47
50
53
54
55
56
57
59
63
64
69
70
72
73
75
82
83
64
85
66
87
88
89
90
91
92
93
94
95
96
97
98
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
266
K. A. PERKO, JR.
99
100
103
109
110
111
112
113
114
115
116
144 3VI
145 3VII
146 3VIII
147 3IX
148 3X
149 41
150 411
151 4III
152 4IV
153 4V
154 4VI
155
156
157
158
159
160
161
162
163
164
165
4VII
4VIII
51
511
6l
611
6111
6IV
6V
6VII
6VIII
REFERENCES 1.
G. Bürde,
Darstellungen
von Knotengruppen
und eine
Abh. Math. Sem. Univ. Hamburg 35 (1970), 107-120. 2. braic
J.H.
Conway,
properties,
An enumeration
Computational
of knots
Problems
and links,
in Abstract
R. H. Fox,
(1970), 193-201.
Metacyclic
invariants
of knots
and some
/\lgebra
1967), Pergamon Press, Oxford, 1970, pp. 329-358. 3-
Knoteninvariante,
MR 43 #2695of their
(Proc.
Conf.,
algeOxford,
MR 41 #2661. and links,
Canad.
J. Math.
22
MR41#6197.
4. C. N. Little,
Non-alternate
± knots,
Trans.
Roy. Soc. Edinburgh
39 (1900),
771-778, plates I, II, III. 5-
K. Murasugi,
Oti a certain
Math. Soc. 117 (1965), 387-422. 6.
biete,
K. Reidemeister,
numerical
Knotentheorie,
Band 1, Springer-Verlag,
invariant
of link
types,
Trans.
Amer.
MR 30 #1506. Berlin,
Ergebnisse
der Math,
und ihrer
Grenzge-
1932.
7. H. Schubert, Knoten mit zwei Brücken, Math. Z. 65 (1956), 133—170. MR 18, 498. 8.
P. G. Tait,
The first
seven
orders
of knottiness,
Trans.
Roy.
Soc.
Edinburgh
32(1884), plate 44. 9. -,
Tenfold knottiness,
Trans.
Roy. Soc. Edinburgh
32 ( 1885), plates
80, 81. 400 Central Park West Apt. 16-P, New York, New York 10025 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
