#### Volume 15, issue 6 (2015)

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A colored operad for string link infection

### John Burke and Robin Koytcheff

Algebraic & Geometric Topology 15 (2015) 3371–3408
##### Abstract

Budney constructed an operad that encodes splicing of knots and further showed that the space of (long) knots is generated over this splicing operad by the space of torus knots and hyperbolic knots. This generalized the satellite decomposition of knots from isotopy classes to the level of the space of knots. Infection by string links is a generalization of splicing from knots to links. We construct a colored operad that encodes string link infection. We prove that a certain subspace of the space of $2$–component string links is generated over a suboperad of our operad by its subspace of prime links. This generalizes a result from joint work with Blair from isotopy classes of string links to the space of string links. Furthermore, all the relations in the monoid of $2$–string links (as determined in our joint work with Blair) are captured by our infection operad.

##### Keywords
string links, infection, operads, spaces of links, spaces of knots
##### Mathematical Subject Classification 2010
Primary: 57M25, 18D50, 55P48, 57R40, 57R52
##### Publication
Received: 11 August 2014
Revised: 25 February 2015
Accepted: 28 February 2015
Published: 12 January 2016
##### Authors
 John Burke Department of Mathematics Rhode Island College Providence, RI 02908-1991 USA Robin Koytcheff Department of Mathematics and Statistics University of Massachusetts Amherst, MA 01003 USA http://people.math.umass.edu/~koytcheff