Closed incompressible surfaces in knot complements

Finkelstein, Elizabeth; Moriah, Yoav

doi:10.1090/S0002-9947-99-02233-3

Closed incompressible surfaces in knot complements
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by Elizabeth Finkelstein and Yoav Moriah PDF

Trans. Amer. Math. Soc. 352 (2000), 655-677 Request permission

Abstract:

In this paper we show that given a knot or link $K$ in a $2n$-plat projection with $n\ge 3$ and $m\ge 5$, where $m$ is the length of the plat, if the twist coefficients $a_{i,j}$ all satisfy $|a_{i,j}|>1$ then $S^3-N(K)$ has at least $2n-4$ nonisotopic essential meridional planar surfaces. In particular if $K$ is a knot then $S^3-N(K)$ contains closed incompressible surfaces. In this case the closed surfaces remain incompressible after all surgeries except perhaps along a ray of surgery coefficients in $\mathbb {Z}\oplus \mathbb {Z}$.

References

Gerhard Burde and Heiner Zieschang, Knots, De Gruyter Studies in Mathematics, vol. 5, Walter de Gruyter & Co., Berlin, 1985. MR 808776
Marc Culler, C. McA. Gordon, J. Luecke, and Peter B. Shalen, Dehn surgery on knots, Ann. of Math. (2) 125 (1987), no. 2, 237–300. MR 881270, DOI 10.2307/1971311
D. Cooper and D. D. Long, Derivative varieties and the pure braid group, Amer. J. Math. 115 (1993), no. 1, 137–160. MR 1209237, DOI 10.2307/2374725
E. Finkelstein, Closed incompressible surfaces in closed braid complements, J. Knot Theory Ramifications 7 (1998), 335–379.
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
C. McA. Gordon and A. W. Reid, Tangle decompositions of tunnel number one knots and links, J. Knot Theory Ramifications 4 (1995), no. 3, 389–409. MR 1347361, DOI 10.1142/S0218216595000193
John Hempel, $3$-Manifolds, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. MR 0415619
William H. Jaco and Peter B. Shalen, Seifert fibered spaces in $3$-manifolds, Mem. Amer. Math. Soc. 21 (1979), no. 220, viii+192. MR 539411, DOI 10.1090/memo/0220
D. Heath and T. Kobayashi, A search method for a thin position of a link, preprint.
Martin Lustig and Yoav Moriah, Generalized Montesinos knots, tunnels and $\scr N$-torsion, Math. Ann. 295 (1993), no. 1, 167–189. MR 1198847, DOI 10.1007/BF01444882
María Teresa Lozano and Józef H. Przytycki, Incompressible surfaces in the exterior of a closed $3$-braid. I. Surfaces with horizontal boundary components, Math. Proc. Cambridge Philos. Soc. 98 (1985), no. 2, 275–299. MR 795894, DOI 10.1017/S0305004100063465
Herbert C. Lyon, Incompressible surfaces in knot spaces, Trans. Amer. Math. Soc. 157 (1971), 53–62. MR 275412, DOI 10.1090/S0002-9947-1971-0275412-6
W. Menasco, Closed incompressible surfaces in alternating knot and link complements, Topology 23 (1984), no. 1, 37–44. MR 721450, DOI 10.1016/0040-9383(84)90023-5
Ulrich Oertel, Closed incompressible surfaces in complements of star links, Pacific J. Math. 111 (1984), no. 1, 209–230. MR 732067
Hamish Short, Some closed incompressible surfaces in knot complements which survive surgery, Low-dimensional topology (Chelwood Gate, 1982) London Math. Soc. Lecture Note Ser., vol. 95, Cambridge Univ. Press, Cambridge, 1985, pp. 179–194. MR 827302, DOI 10.1017/CBO9780511662744.007
G. Ananda Swarup, On incompressible surfaces in the complements of knots, J. Indian Math. Soc. (N.S.) 37 (1973), 9–24 (1974). MR 362315
Abigail Thompson, Thin position and bridge number for knots in the $3$-sphere, Topology 36 (1997), no. 2, 505–507. MR 1415602, DOI 10.1016/0040-9383(96)00010-9
Ying Qing Wu, Incompressibility of surfaces in surgered $3$-manifolds, Topology 31 (1992), no. 2, 271–279. MR 1167169, DOI 10.1016/0040-9383(92)90020-I
Ying-Qing Wu, The classification of nonsimple algebraic tangles, Math. Ann. 304 (1996), no. 3, 457–480. MR 1375620, DOI 10.1007/BF01446301