A colored operad for string link infection

Burke, John¹ ; Koytcheff, Robin²

¹Department of Mathematics, Rhode Island College, Providence, RI 02908-1991, USA
²Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003, USA

Algebraic and Geometric Topology, Tome 15 (2015) no. 6, pp. 3371-3408

Résumé

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.

Détail

DOI : 10.2140/agt.2015.15.3371

Classification : 57M25, 18D50, 55P48, 57R40, 57R52
Keywords: string links, infection, operads, spaces of links, spaces of knots

Bibliographie

[1] R Blair, J Burke, R Koytcheff, A prime decomposition theorem for the $2$–string link monoid, J. Knot Theory Ramifications 24 (2015)

[2] R Budney, Little cubes and long knots, Topology 46 (2007) 1

[3] R Budney, Topology of knot spaces in dimension $3$, Proc. Lond. Math. Soc. 101 (2010) 477

[4] R Budney, An operad for splicing, J. Topol. 5 (2012) 945

[5] J Cerf, Sur les difféomorphismes de la sphère de dimension trois $(\Gamma _{4}=0)$, Lecture Notes in Mathematics 53, Springer (1968)

[6] D Cimasoni, Slicing Bing doubles, Algebr. Geom. Topol. 6 (2006) 2395

[7] T D Cochran, Noncommutative knot theory, Algebr. Geom. Topol. 4 (2004) 347

[8] T Cochran, S Friedl, P Teichner, New constructions of slice links, Comment. Math. Helv. 84 (2009) 617

[9] T D Cochran, S Harvey, C Leidy, Primary decomposition and the fractal nature of knot concordance, Math. Ann. 351 (2011) 443

[10] T D Cochran, K E Orr, Homology boundary links and Blanchfield forms: Concordance classification and new tangle-theoretic constructions, Topology 33 (1994) 397

[11] T D Cochran, K E Orr, P Teichner, Knot concordance, Whitney towers and $L^2$–signatures, Ann. of Math. 157 (2003) 433

[12] T D Cochran, K E Orr, P Teichner, Structure in the classical knot concordance group, Comment. Math. Helv. 79 (2004) 105

[13] J H Conway, An enumeration of knots and links, and some of their algebraic properties, from: "Computational Problems in Abstract Algebra" (editor J Leech), Pergamon (1970) 329

[14] P M Gilmer, Slice knots in $S^{3}$, Quart. J. Math. Oxford Ser. 34 (1983) 305

[15] K Habiro, Claspers and finite type invariants of links, Geom. Topol. 4 (2000) 1

[16] S L Harvey, Homology cobordism invariants and the Cochran–Orr–Teichner filtration of the link concordance group, Geom. Topol. 12 (2008) 387

[17] A Hatcher, Homeomorphisms of sufficiently large $P^{2}$–irreducible $3$–manifolds, Topology 15 (1976) 343

[18] A E Hatcher, A proof of the Smale conjecture, $\mathrm{Diff}(S^{3})\simeq \mathrm{O}(4)$, Ann. of Math. 117 (1983) 553

[19] M W Hirsch, Differential topology, Graduate Texts in Mathematics 33, Springer (1976)

[20] V F R Jones, Planar algebras, I, preprint (1999)

[21] J Y Le Dimet, Cobordisme d'enlacements de disques, Mém. Soc. Math. France 32, Soc. Math. France (1988)

[22] C Livingston, A survey of classical knot concordance, from: "Handbook of knot theory" (editors W Menasco, M Thistlethwaite), Elsevier (2005) 319

[23] M Markl, S Shnider, J Stasheff, Operads in algebra, topology and physics, Mathematical Surveys and Monographs 96, Amer. Math. Soc. (2002)

[24] J P May, The geometry of iterated loop spaces, Lecture Notes in Mathematics 271, Springer (1972)

[25] J E Mcclure, J H Smith, Operads and cosimplicial objects: An introduction, from: "Axiomatic, enriched and motivic homotopy theory" (editor J P C Greenlees), NATO Sci. Ser. II Math. Phys. Chem. 131, Kluwer (2004) 133