科研之友
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摘要
A new topological operad is introduced, called the splicing operad. This operad acts on a broad class of spaces of self-embeddings N -%26gt; N, where N is a manifold. The action of this operad on EC(j, M) (self-embeddings R-j x M. R-j x M with support in I-j x M) is an extension of the action of the operad of (j + 1)-cubes on this space defined in Budney [%26apos;Little cubes and long knots%26apos;, Topology 46 (2007) 1-27]. Moreover, the action of the splicing operad encodes a version of Larry Siebenmann%26apos;s [Bonahon and Siebenmann, %26apos;New Geometric Splittings of Classical Knots, and the Classification and Symmetries of Arborescent Knots%26apos;, Preprint; Siebenmann, %26apos;On vanishing of the Rohlin invariant and nonfinitely amphicheiral homology 3-spheres%26apos;, Proc. Sympos., Univ. Siegen, Siegen, 1979, Lecture Notes in Math. 788 (Springer, Berlin, 1980) 172222] splicing construction for knots in S-3 in the j = 1, M = D-2 case, for which we denote the splicing operad SP3,1. The space of long knots in R-3 (denoted by K-3,K-1) was shown to be a free algebra over the 2-cubes operad with free generating subspace P subset of K-3,K-1, the subspace of long knots that are prime with respect to the connect-sum operation [R. Budney, %26apos;Little cubes and long knots%26apos;, Topology 46 (2007) 1-27]. One of the main results of this paper is that K-3,K-1 is free with respect to the splicing operad SP3,1 action, but the free generating space is the significantly smaller space of torus and hyperbolic knots T H subset of K-3,K-1. Moreover, we show that SP3,1 is a free product of two operads. The first free summand of SP3,1 is a semi-direct product C-2 x O-2 operad which is not equivalent to the framed discs operad. The second free summand of SP3,1 is a free Sigma (sic) O-2-operad, free on Sigma (sic) O-2-spaces that encode cabling and hyperbolic satellite operations; moreover, the Sigma (sic) O-2-homotopy type of these spaces is determined by finding adapted maximal symmetry positions for hyperbolic links in S-3. This is an in-principle explicit description of the homotopy type of the space of knots in S-3, and modulo the rather difficult problem of determining the symmetry groups of a class of hyperbolic links and their actions on the cusps, this is a closed form of description of the homotopy type.
- 出版日期2012
