# zbMATH — the first resource for mathematics

Tangles, property P, and a problem of J. Martin. (English) Zbl 0563.57002
A band on a link $$L\subset S^ 3$$ is an embedding $$b: I\times I\to S^ 3$$ such that $$b^{-1}(L)=\partial I\times I$$. A band b is trivial if there is a disk in $$S^ 3$$ whose interior is disjoint from $$L\cup image(b)$$ and whose boundary is the union of b(I$$\times \{0\})$$ and a subarc of L. Say $$L_ n$$ is obtained by an n-twist banding of L if $$L_ n$$ is obtained by replacing the two arcs b($$\partial I\times I)$$ of a band b on L with a pair of arcs near b.
Theorem: If K and $$K_ n$$ are both the unknot, and $$| n| \geq 2$$, then the band is trivial. In the notation of the Conway calculus we have, using $$n=2:$$ Corollary 1: If $$L_+$$ and $$L_-$$ are the unknot, $$L_ 0$$ is the unlink. Corollary 2: An amphicheiral, strongly invertible knot has property P. Corollary 1 solves problem 1.18 on the 1978 Kirby problem list. The case $$| n| =1$$ is far more difficult, but true, and is the subject of a later paper. Its solution removes ”amphicheiral” from Corollary 2.

##### MSC:
 57M25 Knots and links in the $$3$$-sphere (MSC2010)
band on a link
Full Text:
##### References:
  Bass, H., Birman, J., Morgan, J.: The Smith conjecture. London, New York: Academic Press 1984 · Zbl 0599.57001  Bleiler, S., Scharlemann, M.: Aprojective plane in ?4 with three critical points is standard. MSRI] preprint · Zbl 0678.57003  Bleiler, S.: Strongly invertible knots have propertyR Math. Z. (to appear) · Zbl 0577.57001  Bleiler, S.: Prime tangles and composite knots. Proceedings vancouver 1983. Lect. Notes Math. Berlin, Heidelberg, New York, Tokyo: Springer (to appear) · Zbl 0596.57003  Culler, M., Gordon, C., Luecke, J., Shaler, P.: Dehn surgery on knots. MSRI preprint  Furasawa, F., Sakuma, M.: Dehn surgery on symmetric knots. Math. Semin. Notes, Kobe Univ.11, 179-198 (1983) · Zbl 0576.57002  Hodgson, C.: Involutions and isotopies of Lens spaces. Master’s thesis University of Melbourne 1981  Kirby, R.: Problems in low-dimensional topology. Proc. Symp. Pure Math.32, 273-312 (1978)  Marumoto, Y.: Relations between some conjectures in knot theory. Math. Semin. Notes, Kobe Univ.5, 377-388 (1977) · Zbl 0374.55001  Scharlemann, M.: Smooth spheres in ?4 with four critical points are standard. Invent. Math.79, 125-141 (1985) · Zbl 0559.57019 · doi:10.1007/BF01388659  Scharlemann, M.: Unknotting number one knots are prime. Invent. Math.82, 37-55 (1985) · Zbl 0576.57004 · doi:10.1007/BF01394778  Waldhausen, F.: Über Involutionen der 3-Sphäre, Topology8, 81-91 (1969) · Zbl 0185.27603 · doi:10.1016/0040-9383(69)90033-0
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.