On the classification of knots

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 (1970). Primary 55A25, 55A10. 12, 1973. 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 https://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 their grouped distinguish 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 https://www.ams.org/journal-terms-of-use in or, in 3 cases, (and corresponding, table. Figure all but 31 have and from the knots polynomials which constitute forth these numbers tables, 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 ±6/5 820 ±2 10, ±4 ±4 ±6 ±6 10, ±6/5 10, 928 ±4 929 ±10 103VJ ±2 ±6 ±4/5 ±4/5 ±4/5 ±4/5 ±4/5 +4/5 104 ±■1 ±4 ±4 ±6 10« ±2 10, 1046 I0„, ±4 0 ±4 102 ±10 10.,v 10, ±4 ±4 ±4 U4/5 8,6 ±12 10, 10, 10, ±12 10, ±24/5 ±24/5 ±24/5 ±24/5 ±24/5 +56/5 ±10/3 114/3 ±14/3 10, ho 'O4V,,, +4 +12 ±2 10, 93a ±14 10.„ u69 ±4 0 »o« 10, 10, 1077 940 « 1042 ±2 102 "29(2.-2) 0 10, ±4 ±52/11 su «2/11 6VI «2/11 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 10, 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 https://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 4VII 4VIII 157 158 51 159 511 160 6l 161 611 162 6111 163 6IV 164 6V 165 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 https://www.ams.org/journal-terms-of-use