The Alexander and Markov theorems via diagrams for links in $3$-manifolds
HTML articles powered by AMS MathViewer
- by Paul A. Sundheim PDF
- Trans. Amer. Math. Soc. 337 (1993), 591-607 Request permission
Abstract:
Let $M$ be a $3$-manifold with an open book decomposition. We obtain a new proof that a link in $M$ has a braided form and that two braided forms are related by a sequence of two Markov moves for $M$ by generalizing Morton’s approach for links in ${S^3}$.References
-
J. W. Alexander, A lemma on systems of knotted curves, Proc. Acad. Sci. U.S.A. 9 (1923), 93-95.
- Israel Berstein and Allan L. Edmonds, On the construction of branched coverings of low-dimensional manifolds, Trans. Amer. Math. Soc. 247 (1979), 87–124. MR 517687, DOI 10.1090/S0002-9947-1979-0517687-9
- Joan S. Birman, Braids, links, and mapping class groups, Annals of Mathematics Studies, No. 82, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974. MR 0375281 J. Cerf, La stratification naturelle des espaces de fonctions differentiables reeles et la theoreme de la pseudo-isotopy, Publ. Math. I.H.E.S. 39 (1970).
- F. A. Garside, The braid group and other groups, Quart. J. Math. Oxford Ser. (2) 20 (1969), 235–254. MR 248801, DOI 10.1093/qmath/20.1.235
- V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), no. 2, 335–388. MR 908150, DOI 10.2307/1971403 A. A. Markov, Über die freie Aquivalenz der geschlossner Zopfe, Rec. Soc. Math. Moscou 1 (1935), 73-78.
- H. R. Morton, Threading knot diagrams, Math. Proc. Cambridge Philos. Soc. 99 (1986), no. 2, 247–260. MR 817666, DOI 10.1017/S0305004100064161 R. Myres, Open book decompositions of $3$-manifolds, Proc. Amer. Math. Soc. 72 (1978). K. Reidemeister, Knotentheorie, Chelsea, New York, 1948.
- Lee Rudolph, Constructions of quasipositive knots and links. I, Knots, braids and singularities (Plans-sur-Bex, 1982) Monogr. Enseign. Math., vol. 31, Enseignement Math., Geneva, 1983, pp. 233–245. MR 728589 R. K. Skora, Braids in $3$-manifolds, (1988).
- Paul A. Sundheim, Reidemeister’s theorem for $3$-manifolds, Math. Proc. Cambridge Philos. Soc. 110 (1991), no. 2, 281–292. MR 1113426, DOI 10.1017/S0305004100070353 E. Witten, Quantum field theory and the Jones polynomial, preprint.
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 337 (1993), 591-607
- MSC: Primary 57M25; Secondary 57M50
- DOI: https://doi.org/10.1090/S0002-9947-1993-1179401-3
- MathSciNet review: 1179401