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• W2003297974 abstract "This paper gives a method for constructing all links in S3, beginning with the unknot and adding at most one to the norm of the link at each stage. This has two corollaries. The first is that links with 'minimal' skein trees are fibered. The second is a complete list of all links with skein trees of height two. In [2] it is shown that three links related by the Conway moves have closely related Thurston norm minimizing Seifert surfaces. This is used to give a lower bound on the height of a skein tree for a link. Here these results are extended to yield a method for constructing all links, beginning with the unknot and adding at most one to the norm of the link at each stage. This has two corollaries: the first is that if the lower bound obtained in [2] is realized, then the link must be fibered. The second is a complete list of all links with skein trees of height two. It is surprisingly easy to obtain this second corollary, especially when one considers that the question of characterizing skein trees of height one is equivalent to the question of whether one can band together two knots in some non-trivial way to obtain the unknot. This was a long-standing problem, eventually solved by M. Scharlemann [1]. (1.1 ) Definition. A Seifert surface for an oriented link L in S3 is an oriented surface S, with no closed components, such that aS = L. S is taut if there is no Seifert surface for L of larger Euler characteristic. This is equivalent to being Thurston norm minimizing and incompressible [2]. Let X(L) be minus the Euler characteristic of a taut Seifert surface for L. (1.2) Definition. Three oriented links L+, L_ and Lo in S3 are related by the Conway moves if they are identical outside a three-ball B where they appear as in Figure 1. We refer to [2] for the definitions of a skein tree T for a link L, a node of T, a leaf of T, the height of a skein tree for L and the height of L, written h(L). Received by the editors August 25, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 57M25. The author was supported in part by a National Science Foundation grant and the Lady Davis Fellowship Trust. i) 1989 American Mathematical Society 0002-9939/89 $1.00 + $.25 per page" @default.
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• W2003297974 date "1989-04-01" @default.
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• W2003297974 title "Thurston norm minimizing surfaces and skein trees for links in $Ssp 3$" @default.
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• W2003297974 doi "https://doi.org/10.1090/s0002-9939-1989-0969321-3" @default.
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