Networking Seifert surgeries on knots (Q2882488)

In this memoir the authors take a new approach in studying integral Dehn surgeries on knots in \(S^3\) yielding Seifert fibered spaces. The approach is to find the network between integral Seifert surgeries. A Seifert surgery on a knot \(K\) is a surgery on \(K\) that yields a Seifert fibered space or a manifold that admits a degenerate Seifert fibration which contains exceptional fibers of index 0. The network is based on the notion of seiferter for a Seifert surgery on a knot \(K\): this is an unknot \(c\) in \(S^3\) lying in \(S^3- N(K)\) which becomes a regular or exceptional fiber in the resulting Seifert fibered surgery manifold that may contain a degenerate fiber. Two Seifert surgeries are related in the network by twisting operations on a seiferter of one of the surgeries, or by a pair of seiferters that co-bound an annulus.NEWLINENEWLINEIn Chapter 2, an \(S\)-relation is defined between two Seifert surgeries, which is used to define a Seifert Surgery Network. Two Seifert surgeries are \(S\)-related by a seiferter \(c\) or a pair of seiferters \((c_1,c_2)\) if either (1) one surgery has a seiferter \(c\) and the other has a seiferter which is obtained from 1-twist of \(c\), or (2) one surgery has a seiferter pair \((c_1,c_2)\) and the other has a seiferter pair which is obtained from 1-twist of \((c_1,c_2)\). For an integer \(m\), an \(m\)-move is defined to transform a knot \(c\) in \(S^3 - N(K)\) to another knot \(c'\) in \(S^3 - N(K)\). It is shown that \(c\) and \(c'\) are \(m\)-equivalent if and only if \(c\) and \(c'\) are isotopic in the resulting surgery manifold with integral slope \(m\); and \(c\) is a seiferter in the surgery manifold with integral slope \(m\) if and only if \(c'\) is. Similarly an \(m\)-move for pairs of seiferters that each co-bound an annulus is defined, and it is shown that a similar result holds for any two pairs.NEWLINENEWLINEIn Chapter 3, a classification theorem is given for the seiferters of non-degenerate and degenerated Seifert fibrations of a Seifert surgery manifold. It is shown that if a Seifert fibered surgery is on a hyperbolic knot with a seiferter \(c\), then either \(c\) is a hyperbolic seiferter for the surgery, or the surgery manifold has a unique Seifert fibration with \(c\) as a fiber.NEWLINENEWLINEIn Chapter 4, geometric aspects of seiferters are given for hyperbolic knots. A seiferter \(c\) is a geodesic seiferter for a hyperbolic knot \(K\) if \(c\) is a closed geodesic in \(S^3 - K\). It is known that if \(c\) is a closed geodesic in \(S^3 - K\), then \(S^3 - (K \cup c)\) is hyperbolic. Using Thurston's hyperbolic Dehn surgery theorem, a partial converse of the above result is obtained. It is also proved that for any \(r>0\), there exits a hyperbolic seiferter \(c\) and a nontrivial torus knot \(K\) such that \(volume(S^3 - (K \cup c)) > r\).NEWLINENEWLINEChapter 5 gives a global framework of \(S\)-linear trees. A Seifert Surgery Network is obtained by twisting a Seifert surgery successively along a seiferter or an annular pair of seiferters, this \(S\)-family of 1-dimensional sub-complexes is not a linear tree in general. It is proved, however, that the subnetwork generated by a single seiferter is a linear tree (called \(S\)-linear tree). A classification theorem is given for the types of \(S\)-linear trees for a Seifert surgery which is either a lens surgery, a connected sum of two lens space surgeries, or a small Seifert surgery with seiferter \(c\). In the case of hyperbolic seiferters, it is shown that all but at most 4 vertices on the \(S\)-linear tree are Seifert surgeries on hyperbolic knots.NEWLINENEWLINEChapter 6 gives a discussion of the combinatorial structure of the Seifert Surgery Network. It is proved that any two \(S\)-linear trees intersect in at most finitely many vertices; the Seifert Surgery Network contains infinitely many cycles whose vertices are Seifert surgeries on hyperbolic knots; the Network contains a loop edge, which corresponds to an annular pair of seiferters; and the Network is locally finite, namely every Seifert surgery has only finitely many seiferters.NEWLINENEWLINEThe main theorem of Chapter 7 is to answer the question whether a knot can be embedded in a genus 2 Heegaard surface in \(S^3\) if the knot admits a Seifert fibered surgery. In fact, it is shown that there is an infinite family of hyperbolic knots each of which admits a small Seifert fibered surgery, has no symmetry, and cannot be embedded in a genus 2 Heegaard surface in \(S^3\).NEWLINENEWLINEIn Chapter 8, it is shown that the subnetwork whose vertices and edges correspond to the integral surgeries on torus knots and their basic seifeters respectively, is connected. It is also shown that the subnetwork whose vertices and edges correspond to the Seifert surgeries on graph knots and the basic seiferters for their companion torus knot, is connected.NEWLINENEWLINEIn the final Chapter 9, paths from various examples of Seifert surgeries to those on torus knots are demonstrated. These examples consist of Seifert surgeries obtained by Berges' construction producing lens spaces, Dean's Seifert fibered surgeries, and Seifert surgeries by the Montesinos trick. Examples of toroidal integral Seifert surgeries connected to the subnetwork of torus knots are also given.