Unlinking via simultaneous crossing changes

Scharlemann, Martin

doi:10.1090/S0002-9947-1993-1200011-3

Unlinking via simultaneous crossing changes
HTML articles powered by AMS MathViewer

by Martin Scharlemann PDF

Trans. Amer. Math. Soc. 336 (1993), 855-868 Request permission

Abstract:

Given two distinct crossings of a knot or link projection, we consider the question: Under what conditions can we obtain the unlink by changing both crossings simultaneously? More generally, for which simultaneous twistings at the crossings is the genus reduced? Though several examples show that the answer must be complicated, they also suggest the correct necessary conditions on the twisting numbers.

References

John W. Morgan and Hyman Bass (eds.), The Smith conjecture, Pure and Applied Mathematics, vol. 112, Academic Press, Inc., Orlando, FL, 1984. Papers presented at the symposium held at Columbia University, New York, 1979. MR 758459
Steven Bleiler and Martin Scharlemann, Tangles, property $P$, and a problem of J. Martin, Math. Ann. 273 (1986), no. 2, 215–225. MR 817877, DOI 10.1007/BF01451402

Involutions et fibrés de Seifert dans les variétés de dimension

Gerhard Burde and Heiner Zieschang, Knots, De Gruyter Studies in Mathematics, vol. 5, Walter de Gruyter & Co., Berlin, 1985. MR 808776
David Gabai, Foliations and the topology of $3$-manifolds, J. Differential Geom. 18 (1983), no. 3, 445–503. MR 723813
C. McA. Gordon and J. Luecke, Reducible manifolds and Dehn surgery, Topology 35 (1996), no. 2, 385–409. MR 1380506, DOI 10.1016/0040-9383(95)00016-X
W. B. Raymond Lickorish, The unknotting number of a classical knot, Combinatorial methods in topology and algebraic geometry (Rochester, N.Y., 1982) Contemp. Math., vol. 44, Amer. Math. Soc., Providence, RI, 1985, pp. 117–121. MR 813107, DOI 10.1090/conm/044/813107
José M. Montesinos, Surgery on links and double branched covers of $S^{3}$, Knots, groups, and $3$-manifolds (Papers dedicated to the memory of R. H. Fox), Ann. of Math. Studies, No. 84, Princeton Univ. Press, Princeton, N.J., 1975, pp. 227–259. MR 0380802
Martin Scharlemann, Sutured manifolds and generalized Thurston norms, J. Differential Geom. 29 (1989), no. 3, 557–614. MR 992331
Martin Scharlemann, Handlebody complements in the $3$-sphere: a remark on a theorem of Fox, Proc. Amer. Math. Soc. 115 (1992), no. 4, 1115–1117. MR 1116272, DOI 10.1090/S0002-9939-1992-1116272-X
Martin Scharlemann, Producing reducible $3$-manifolds by surgery on a knot, Topology 29 (1990), no. 4, 481–500. MR 1071370, DOI 10.1016/0040-9383(90)90017-E
Martin Scharlemann and Abigail Thompson, Link genus and the Conway moves, Comment. Math. Helv. 64 (1989), no. 4, 527–535. MR 1022995, DOI 10.1007/BF02564693
William P. Thurston, A norm for the homology of $3$-manifolds, Mem. Amer. Math. Soc. 59 (1986), no. 339, i–vi and 99–130. MR 823443