Unknotting number, genus and companion tori. (English) Zbl 0616.57003

We apply the Gabai sutured manifold machine to study the effect of crossing changes on the genus of a knot. Our results are of two types:
First, we show that there are knots for which a crossing change which lowers the unknotting number raises the genus. In fact the argument seems, in principle, to produce a myriad of such examples.
Second, we examine the effect of companion tori on crossing changes. We discover that either such companion tori are essentially unrelated to the part of the knot on which the crossing change is being made, or there is an inequality which relates the genus of the knot before and after the crossing change.
The precise statement and some corollaries are:
Definition: Let K be a knot in \(S^ 3\) and D be a disk which intersects K in precisely two points, of opposite orientation. Let \(L=\partial D\) be given a \(\pm 1\)-framing. Then L is a crossing link and D is its crossing disk. Denote by K’ the knot in \(S^ 3\) obtained from K by doing \(\pm 1\) surgery to L. (K and K’ are related by a single crossing change.) Let T be an incompressible torus in \(S^ 3-K.\)
Theorem: If genus(K’)\(\leq genus(K)-2\) then any companion torus of K can be isotoped disjoint from L.
Corollary: If K is a satellite knot of winding number \(\neq 0\), then the unknotting number of K is greater than one.
Corollary: An unknotting number one knot is prime.


57M25 Knots and links in the \(3\)-sphere (MSC2010)
Full Text: DOI EuDML


[1] Bleiler, S.: Prime tangles and composite knots. In: Knot theory and manifolds. Lecture Notes 1144, pp. 1-13. Berlin, Heidelberg, New York: Springer 1985
[2] Eudave-Muñoz, M.: Cirugia en nudos fuertemente invertibles. An. Inst. Mat. U.N.A.M. 26, 41-57 (1986)
[3] Gabai, D.: Foliations and the topology of 3-manifolds, II. J. Differ. Geom.26, 461-478 (1987) · Zbl 0627.57012
[4] Gabai, D.: Genus is superadditive under band-connected sum. Topology26, 209-210 (1987) · Zbl 0621.57004
[5] Gordon, C., Luecke, J.: Only integral Dehn surgeries can yield reducible manifolds. Math. Proc. Cam. Phil. Soc.102, 97-101 (1987) · Zbl 0655.57500
[6] Lyon, H.: Simple knots without unique minimal surfaces. Proc. Am. Math. Soc.43, 449-454 (1974) · Zbl 0247.55003
[7] Myers, R.: Simple knots in compact, orientable 3-manifolds. Trans. Am. Math. Soc.273, 75-91 (1982) · Zbl 0508.57008
[8] Scharlemann, M.: Smooth spheres in ?4 with four critical points are standard. Invent. Math.79, 125-141 (1985) · Zbl 0559.57019
[9] Scharlemann, M.: Unknotting number one knots are prime. Invent. Math.82, 37-55 (1985) · Zbl 0576.57004
[10] Scharlemann, M.: Sutured manifolds and generalized Thurston norms. To appear in J. Differ. Geom. · Zbl 0673.57015
[11] Thompson, A.: PropertyP for the band-connect sum of two knots. Topology26, 205-208 (1987) · Zbl 0628.57005
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.