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**Links, bridge number, and width trees.**
*(English)*
Zbl 1520.57003

Height functions for a link in the sphere \(S^3\) and bridge surfaces, consisting of thin and thick spheres and associated with these functions, allow to define the bridge number, the Gabai width and the other geometric invariants of knots and links. By modifying the bridge position for a link and endowing the bridge surfaces with an additional structure, several authors introduced new geometric knot invariants which generalize the bridge number and the Gabai width (see, for example [R. Blair et al., Algebr. Geom. Topol. 13, No. 5, 2925–2946 (2013; Zbl 1278.57004)]). In particular the “net extent” and “width” generalize the classical knot invariants mentioned above and produce additional topological insight into them [S. A. Taylor and M. Tomova, Geom. Topol. 22, No. 6, 3235–3286 (2018; Zbl 1400.57012)]. The notions of multiple bridge surface for a link or a tangle and the width trees associated with the multiple bridge surface play a key role in construction of these new invariants.

In the paper under review, the authors introduce and study multiple \(c\)-bridge surfaces for tangles and links. With every even tangle \((M,\tau)\), including each knot or link in the \(3\)-sphere, they associate a collection \(\mathbf{T}_2 (M,\tau )\) of width trees. A width tree is a purely combinatorial object represented by a digraph and endowed with additional marks and is considered up to a certain equivalence relation. The authors define and study “netextent” and “width” of width trees, the combinatorial invariants corresponding to knot invariants defined in [loc. cit.]. By using these new objects, the authors give a graph theoretic reformulation of the generalized bridge position for links in the \(3\)-sphere. As a consequence, it is shown that such knot invariants as the bridge number, the Gabai width and the Taylor-Tomova invariants can be computed by minimizing over the elements of \(\textbf{T}_2 (M,\tau )\). Moreover in the case of productless positive width trees, specified by the authors, the combinatorial invariants “netextent” and “width” of trees \((T, \lambda)\) can be realized by the values of the corresponding topological invariants “netextent” and “width” of knots \(K\) in \(S^3\) such that \((S^3, K)\) is associated with \((T, \lambda)\).

In the paper under review, the authors introduce and study multiple \(c\)-bridge surfaces for tangles and links. With every even tangle \((M,\tau)\), including each knot or link in the \(3\)-sphere, they associate a collection \(\mathbf{T}_2 (M,\tau )\) of width trees. A width tree is a purely combinatorial object represented by a digraph and endowed with additional marks and is considered up to a certain equivalence relation. The authors define and study “netextent” and “width” of width trees, the combinatorial invariants corresponding to knot invariants defined in [loc. cit.]. By using these new objects, the authors give a graph theoretic reformulation of the generalized bridge position for links in the \(3\)-sphere. As a consequence, it is shown that such knot invariants as the bridge number, the Gabai width and the Taylor-Tomova invariants can be computed by minimizing over the elements of \(\textbf{T}_2 (M,\tau )\). Moreover in the case of productless positive width trees, specified by the authors, the combinatorial invariants “netextent” and “width” of trees \((T, \lambda)\) can be realized by the values of the corresponding topological invariants “netextent” and “width” of knots \(K\) in \(S^3\) such that \((S^3, K)\) is associated with \((T, \lambda)\).

Reviewer: Leonid Plachta (Kraków)

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