×

Uniqueness of symmetries of knots. (English) Zbl 0578.57003

A symmetry of a knot K in \(S^ 3\) is a finite (cyclic) group action on \((S^ 3,K)\) which preserves the orientations of both \(S^ 3\) and K. The following problems are considered: (a) Are the symmetries of K of a given order unique? (b) More strongly, does K admit a ”universal symmetry”? The author gives certain sufficient conditions for (a) and (b) to hold. Further by examples, the conditions are seen to be the ”best possible”. It should be noted that there is a knot, for which (a) is affirmative but (b) is negative.
For semi-free symmetries, the following results are obtained: (1) A free symmetry of a prime knot is uniquely determined by its order. (2) A symmetry of a knot which realizes a cyclic period n is unique, if \(n\geq 3\). (3) For any integer s, there is a knot which admits more than s inequivalent symmetries which realize cyclic period 2. The result (1) was also obtained by M. Boileau and E. Flapan [preprint], independently.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57S17 Finite transformation groups
57S25 Groups acting on specific manifolds
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Boileau, C.M.: Noeuds rigidement inversibles. Low dimensional topology, London Math. Soc. Lect. Note Series 95, pp. 1-18. Cambridge Univ. Press 1985 · Zbl 0571.57006
[2] Boileau, C.M., Flapan, E.: Uniqueness of free actions onS 3 respecting a knot. Preprint · Zbl 0637.57006
[3] Bonahon, M., Siebenmann, L.: Seifert 3-orbifolds and their role as natural crystalline parts of arbitrary compact irreducible 3-orbifolds. To appear in L.M.S. lecture note series. · Zbl 0571.57011
[4] Bredon, G.: Introduction to compact transformation groups. Pure Appl. Math. Sci.46. New York: Academic Press 1972 · Zbl 0246.57017
[5] Burde, G., Murasugi, K.: Links and Seifert fiberd spaces. Duke Math. J.37, 89-93 (1970) · Zbl 0195.54003 · doi:10.1215/S0012-7094-70-03713-0
[6] Edmonds, A.: Least area Seifert surfaces and periodic knots. Topology Appl.18, 109-113 (1984) · Zbl 0557.57003 · doi:10.1016/0166-8641(84)90003-8
[7] Edmonds, A., Livingston, C.: Group actions on fibered 3-manifolds. Comment. Math. Helv.58, 529-542 (1983) · Zbl 0532.57024 · doi:10.1007/BF02564651
[8] Flapan, E.: Infinitely periodic knots, Can. J. Math.37, 17-28 (1985) · Zbl 0571.57007 · doi:10.4153/CJM-1985-002-4
[9] Flapan, E.: Companions of periodic knots are periodic. Preprint · Zbl 0571.57007
[10] Hartley, R.: Knots and involutions. Math. Z.171, 175-185 (1980) · Zbl 0423.57003 · doi:10.1007/BF01176707
[11] Hartley, R.: Knots with free periods. Can. J. Math.33, 91-102 (1981) · Zbl 0481.57003 · doi:10.4153/CJM-1981-009-7
[12] Hillman, J.A.: Links with infinitely many semifree periods are trivial. Arch. Math.42, 568-572 (1984) · Zbl 0527.57001 · doi:10.1007/BF01194056
[13] Hillman, J.A.: Symmetries of knots and links, and invariants of abelian coverings. Preprint · Zbl 0624.57005
[14] Jaco, W.: Lectures on three manifold topology. Conference board of Math. Sci., Regional Conference Series in Math.43, 1980 · Zbl 0433.57001
[15] Johannson, K.: Homotopy equivalences of 3-manifolds with boundaries. Lect. Notes in Math.761. Berlin Heidelberg New York: Springer 1979 · Zbl 0412.57007
[16] Kojima, S.: Finiteness of symmetries on 3-manifolds. Preprint.
[17] Meeks III, W.H., Scott, P.: Finite group actions on 3-manifolds. Preprint · Zbl 0626.57006
[18] Morgan, J.W., Bass, H. (editor): The Smith conjecture. Pure and Applied Math.112. New York: Academic Press 1984 · Zbl 0599.57001
[19] Myers, R.: Simple knots in compact, orientable 3-manifolds. Trans. A.M.S.273, 75-91 (1982) · Zbl 0508.57008
[20] Nakanishi, Y.: Primeness of links. Math. Sem. Notes, Kobe Univ.9, 415-440 (1981) · Zbl 0491.57003
[21] Nakanishi, Y.: Prime and simple links. Math. Sem. Notes, Kobe Univ.11, 249-256 (1983) · Zbl 0592.57003
[22] Sakuma, M.: Periods of composite links. Math. Sem. Notes, Kobe Univ.9, 445-452 (1981) · Zbl 0484.57003
[23] Sakuma, M.: Involutions on torus bundles overS 1. Osaka J. Math.22, 163-185 (1985) · Zbl 0571.57028
[24] Sakuma, M.: On strongly invertible knots. To appear in the Miyata Memorial Symposium (held at Kinosaki, Japan, 1984), Kinokuniya-Shoten
[25] Sakuma, M.: Non-free-periodicity of amphicheiral hyperbolic knots. To appear in the proceeding of the topology symposium held at Kyoto 1984. North-Holland, Amsterdam
[26] Soma, T.: Simple links and tangles. Tokyo J. Math.6, 65-73 (1983) · Zbl 0531.57006 · doi:10.3836/tjm/1270214326
[27] Thurston, W.P.: The geometry and the topology of 3-manifolds. To appear in Lect. Note Series, Princeton Univ. Press
[28] Thurston, W.P.: Three dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. A.M.S.6, 357-381 (1982) · Zbl 0496.57005 · doi:10.1090/S0273-0979-1982-15003-0
[29] Thurston, W.P.: Three manifolds with symmetry. Preprint · Zbl 0324.53031
[30] Tollefson, J.L.: A 3-manifold admitting a unique periodic PL map. Mich. Math. J.21, 7-12 (1974) · Zbl 0291.57003 · doi:10.1307/mmj/1029001202
[31] Waldhausen, F.: On irreducible 3-manifolds which are sufficiently large. Ann. Math.87, 56-88 (1968) · Zbl 0157.30603 · doi:10.2307/1970594
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.