Celebratio Mathematica — Scharlemann

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Works connected to Steven Albert Bleiler

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S. Bleiler and M. Scharlemann: “Tangles, property \( P \), and a problem of J. Martin,” Math. Ann. 273 : 2 (June 1986), pp. 215–225. MR 817877 Zbl 0563.57002 article

In [Scharlemann 1985a; 1985b] new combinatorial techniques are introduced to study the topology of planar surfaces in the complement of genus 2 handlebodies. Here we apply these techniques to two further problems. First, in an argument combining elements of [Scharlemann 1985a; 1985b], we solve problem 1.18B of [Kirby 1978]. The solution is obtained by precisely determining when a pair of unknots can arise during a single “move” in the Conway calculus. Secondly, we show that there is at most one non-trivial Dehn (i.e. rational) surgery on a strongly invertible knot that produces a simply-connected 3-manifold and that such a surgery must have coefficient \( \pm 1 \). This result has since been extended to all knots by Culler et al. [1987] using much less elementary methods. In a related, but combinatorially much more difficult paper [Bleiler and Scharlemann 1988] we solve problem 1.2B of [Kirby 1978] thereby eliminating even the coefficients \( \pm 1 \), and proving property \( P \) for strongly invertible knots.

S. Bleiler and M. Scharlemann: “A projective plane in \( \mathbb{R}^4 \) with three critical points is standard. Strongly invertible knots have property \( P \),” Topology 27 : 4 (1988), pp. 519–540. MR 976593 Zbl 0678.57003 article

Let \( \mathbb{P} \) be the projective plane in \( \mathbb{R}^4 \) obtained by capping off the boundary of an unknotted Möbius band in \( \mathbb{R}^3\times\{0\} \) with an unknotted disk in \( \mathbb{R}^3\times [0,\infty) \). Here we show that any smoothly imbedded projective plane in \( \mathbb{R}^4 \) on which some projection \( \mathbb{R}^4\to\mathbb{R} \) has three nondegenerate critical points is isotopic to \( \mathbb{P} \). The proof is based on a combinatorial solution to Problem 1.2B of [Kirby 1978]. In particular, if a band is attached to an unknot so that the result is an unknot, then the band is isotopic to the trivial half-twisted band. One consequence is that strongly invertible knots have property \( P \) (see [Bleiler and Scharlemann 1986]). Together with [Culler et al. 1987], this further implies that pretzel knots (indeed all symmetric knots) have property \( P \).