×

Waist and trunk of knots. (English) Zbl 1220.57002

Let \(K\) be a non-trivial knot in \(S^3\). Let \(F\) be one of the closed incompressible surfaces in \(S^3 - K\) and let \(D\) be a compressing disk for \(F\) in \(S^3\). Assume that \(D\) intersects \(K\) transversely. The minimum number of intersection points of \(D\) and \(K\) over all compressing disks \(D\) for \(F\) in \(S^3\) is called the waist of \(F\). The waist of \(K\) is defined as the maximum number of \(waist(F)\) over all closed incompressible surfaces \(F\) in \(S^3 - K\), and \(waist(K)=0\) for the trivial knot \(K\). It is known that there are several classes of knots with waist one, for example small knots, toroidally alternating knots by C. Adams [Topology 33, No. 2, 353–369 (1994; Zbl 0839.57004)], and algebraically alternating knots by M. Ozawa [Topology Appl. 157, No. 12, 1937–1948 (2010; Zbl 1217.57008)]. Also, the trunk of knots is defined as the minimum over all the set of Morse functions \(h: S^3 \rightarrow \mathcal{R}\) with two critical points such that \(h\) is also a Morse function on \(K\), over the maximum intersection number of \(h^{-1}(t)\) with \(K\) for all \(t\in \mathcal{R}\).
In the paper under review it is shown that for two knots \(K_1, K_2\) the waist of the connected sum is the maximum of the waists of \(K_1\) and \(K_2\) and that if these knots are meridionally small knots, then the trunk of the connected sum is also the maximum of the trunks of \(K_1\) and \(K_2\). Furthermore, for any knot \(K\) the following inequality is proved: \(waist (K) \leq {trunk (K) /3}\). It is also proved that if \(K\) is a knot with waist one then any incompressible and \(\partial\)-incompressible surface properly embedded in the exterior of \(K\) with boundary of finite slope is free, where free means that the surface properly embedded in the knot exterior cuts it into handlebodies.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Adams C., Brock J., Bugbee J., Comar T., Faigin K., Huston A., Joseph A., Pesikoff D.: Almost alternating links. Topol. Appl. 46, 151–165 (1992) · Zbl 0766.57003 · doi:10.1016/0166-8641(92)90130-R
[2] Adams C.: Toroidally alternating knots and links. Topology 33, 353–369 (1994) · Zbl 0839.57004 · doi:10.1016/0040-9383(94)90017-5
[3] Culler M., Gordon C.McA., Luecke J., Shalen P.B.: Dehn surgery on knots. Ann. Math. 125, 237–300 (1987) · Zbl 0633.57006 · doi:10.2307/1971311
[4] Gabai D.: Foliations and the topology of 3-manifolds. III. J. Differ. Geom. 26, 479–536 (1987) · Zbl 0639.57008
[5] Hatcher A., Thurston W.: Incompressible surfaces in 2-bridge knot complements. Invent. Math. 79, 225–246 (1985) · Zbl 0602.57002 · doi:10.1007/BF01388971
[6] Jaco, W.: Lectures on Three Manifold Topology. AMS Conference board of Math. No. 43 (1980) · Zbl 0433.57001
[7] Lozano M.T., Przytycki J.H.: Incompressible surfaces in the exterior of a closed 3-braid I, surfaces with horizontal boundary components. Math. Proc. Camb. Phil. Soc. 98, 275–299 (1985) · Zbl 0574.57003 · doi:10.1017/S0305004100063465
[8] Menasco W.: Closed incompressible surfaces in alternating knot and link complements. Topology 23, 37–44 (1984) · Zbl 0525.57003 · doi:10.1016/0040-9383(84)90023-5
[9] Morimoto, K.: Essential surfaces in the exteriors of torus knots with twists on 2-strands (preprint)
[10] Oertel U.: Closed incompressible surfaces in complements of star links. Pacific J. Math. 111, 209–230 (1984) · Zbl 0549.57004
[11] Ozawa M.: Synchronism of an incompressible non-free Seifert surface for a knot and an algebraically split closed surface in the knot complement. Proc. Amer. Math. Soc. 128(3), 919–922 (2000) · Zbl 0935.57012 · doi:10.1090/S0002-9939-99-05068-6
[12] Ozawa M.: Closed incompressible surfaces of genus two in 3-bridge knot complements. Topol. Appl. 156, 1130–1139 (2009) · Zbl 1166.57004 · doi:10.1016/j.topol.2008.10.005
[13] Ozawa, M.: Rational structure on algebraic tangles and closed incompressible surfaces in the complements of algebraically alternating knots and links. arXiv:0803.1302 · Zbl 1217.57008
[14] Ozawa, M.: Bridge position and the representativity of spatial graphs, arXiv:0909.1162 · Zbl 1253.05062
[15] Schubert H.: Über eine numerische Knoteninvariante. Math. Z. 61, 245–288 (1954) · Zbl 0058.17403 · doi:10.1007/BF01181346
[16] Schultens J.: Additivity of bridge numbers of knots. Math. Proc. Camb. Phil. Soc. 135, 539–544 (2003) · Zbl 1054.57011 · doi:10.1017/S0305004103006832
[17] Schultens J.: Bridge numbers of torus knots. Math. Proc. Camb. Phil. Soc. 143, 621–625 (2007) · Zbl 1136.57003 · doi:10.1017/S0305004107000448
[18] Thompson A.: Thin position and bridge number for knots in the 3-sphere. Topology 36, 505–507 (1997) · Zbl 0867.57009 · doi:10.1016/0040-9383(96)00010-9
[19] Tomova M.: Compressing thin spheres in the complement of a link. Topol. Appl. 153, 2987–2999 (2006) · Zbl 1106.57006 · doi:10.1016/j.topol.2006.01.006
[20] Wu Y.-Q.: Thin position and essential planar surfaces. Proc. Amer. Math. Soc. 132, 3417–3421 (2004) · Zbl 1055.57012 · doi:10.1090/S0002-9939-04-07416-7
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.