A search method for thin positions of links. (English) Zbl 1086.57005

Let \(S^3\) be \(\{\vec x\in\mathbb R^4 :\|\vec x\| = 1\}\) and let \(h\) be the height function given by projecting \(S^3\) onto a coordinate axis. Let \(L\) be a link in \(S^3\) always assumed to be perturbed so that \(h|_L\) is a Morse function. [D. Gabai [J. Differ. Geom. 26, 479–536 (1987; Zbl 0639.57008)] defined a complexity associated with a link \(L\subset S^3\) called the width of \(L\). The width is calculated as follows: let \(c_0 <\cdots <c_n\) be the critical points of \(h|_L\). Pick points \(r_i\) \((i = 1,\dots,n)\) with \(c_{i-1} < r_i < c_i\). The width of \(L\) is the sum: \(\Sigma_i|L\cap h^{-1}(r_i)|\). After isotopy that realizes the minimal width, \(L\) is said to be in thin position. The purpose of the article under review is describing a search method for thin position.
A level \(h^{-1}(r_i)\) is called thin if \(|L\cap h^{-1}(r_i)| <|L\cap h^{-1}(r_{i+1})|\) and \(|L\cap h^{-1}(t_i)| < |L\cap h^{-1}(r_{i-1})|\). A. Thompson [Topology 36, No. 2, 505–507 (1997; Zbl 0867.57009)] showed that if a link in thin position has a thin level, then the result of compressing this level is an essential, meridional, planar surface in the link exterior. In a previous work [Pac. J. Math. 179, No. 1, 101–117 (1997; Zbl 0887.57012)] the authors of the present paper used Thompson’s result to obtain a tangle decomposition of a given link, that is, a decomposition along essential meridional planar surfaces.
In the paper under review, the authors study this decomposition into “simpler” parts. The simpler parts are obtained as follows: let \(P\) be a component of \(S^3\) cut open along the planar surfaces described above. Then \(P\) is a punctured 3-sphere and \(L\) intersects it in a collection of arcs. The authors then crush each boundary component of \(P\) to obtain \(S^3\) and in \(S^3\) they now have an embedded graph, say \(\Gamma\); the vertices of \(\Gamma\) correspond to the boundary components of \(P\) and the edges are \(L\cap P\). The authors show that \(F\) is in bridge position (that is, the maxima of \(F\) are above its minima). They also show that the vertices of \(F\) are all maxima and minima (a maximum corresponds to a boundary component of \(P\) with \(P\) “below” the component) and assign a plus sign to a vertex that is a maximum and a minus sign to a minimum. A graph with signs assigned to its vertices is called signed.
Finally, the authors use this information to search for thin position; they assume a method for finding bridge position of links and signed graphs is known, as well as an algorithm for finding all essential meridional planar surfaces. An outline of the search process is:
(1) Calculate \(b\), the bridge number of \(L\).
(2) Search for all planar, meridional, essential surfaces with at most \(2(b - 1)\) punctures. Distribute signs to the sides of the spheres so that each sphere gets a plus on one side and minus below. All sign possibilities are considered.
(3) For each system of signed surfaces obtained in (2), cut \(S^3\) along the spheres to obtain signed graphs as described above; place each signed graph in bridge position.
(4) Use the above to construct candidate thin positions for \(L\). A candidate with lowest width is an actual thin position.
The authors conclude with an example: they apply the construction above to calculate thin position the the pretzel link P(3,3,3,3) and show that it has exactly 3 thin levels, each an essential 4-times punctured sphere. The width of P(3,3,3,3,3,3) is 48.


57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M99 General low-dimensional topology
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