Celebratio Mathematica

R. Fenn and C. Rourke: “On Kirby’s calculus of links,” Topology 18 : 1 (1979), pp. 1–15. MR 528232 Zbl 0413.57006 article

R. Fenn and C. Rourke: “Nice spines of 3-manifolds,” pp. 31–36 in Topology of low-dimensional manifolds (Proc. Second Sussex Conf.). Lecture Notes in Math. 722. 1979. MR 547451 Zbl 0406.57002 inproceedings

R. Fenn and C. Rourke: “Racks and links in codimension two,” J. Knot Theory Ramifications 1 : 4 (1992), pp. 343–406. MR 1194995 Zbl 0787.57003 article

A rack, which is the algebraic distillation of two of the Reidemeister moves, is a set with a binary operation such that right multiplication is an automorphism. Any codimension two link has a fundamental rack which contains more information than the fundamental group. Racks provide an elegant and complete algebraic framework in which to study links and knots in 3-manifolds, and also for the 3-manifolds themselves. Racks have been studied by several previous authors and have been called a variety of names. In this first paper of a series we consolidate the algebra of racks and show that the fundamental rack is a complete invariant for irreducible framed links in a 3-manifold and for the 3-manifold itself. We give some examples of computable link invariants derived from the fundamental rack and explain the connection of the theory of racks with that of braids.

R. Fenn, C. Rourke, and B. Sanderson: “Trunks and classifying spaces,” Appl. Categ. Struct. 3 : 4 (1995), pp. 321–356. MR 1364012 Zbl 0853.55021 article

Trunks are objects loosely analogous to categories. Like a category, a trunk has vertices and edges (analogous to objects and morphisms), but instead of composition (which can be regarded as given by preferred triangles of morphisms) it has preferred squares of edges. A trunk has a natural cubical nerve, analogous to the simplicial nerve of a category. The classifying space of the trunk is the realisation of this nerve. Trunks are important in the theory of racks. A rack \( X \) gives rise to a trunk \( \mathcal{T} ( X ) \) which has a single vertex and the set \( X \) as set of edges. The rack space \( B X \) of \( X \) is the realisation of the nerve \( N \mathcal{T} ( X ) \) of \( \mathcal{T} ( X ) \). The connection between the nerve of a trunk and the usual (cubical) nerve of a category determines in particular a natural map \( B X \to B As ( X ) \) where \( B As ( X ) \) is the classifying space of the associated group of X. There is an extension to give a classifying space for an augmented rack, which has a natural map to the loop space of the Brown–Higgins classifying space of the associated crossed module [Fenn and Rourke 1992] and [Brown and Higgins 1991]. The theory can be used to define invariants of knots and links since any invariant of the rack space of the fundamental rack of a knot or link is ipso facto an invariant of the knot or link.

Roger Fenn	Related
Brian Joseph Sanderson	Related

R. Fenn and C. Rourke: “Klyachko’s methods and the solution of equations over torsion-free groups,” Enseign. Math. (2) 42 : 1–2 (1996), pp. 49–74. MR 1395041 Zbl 0861.20029 article

R. Fenn, R. Rimányi, and C. Rourke: “The braid-permutation group,” Topology 36 : 1 (1997), pp. 123–135. MR 1410467 Zbl 0861.57010 article

We consider the subgroup of the automorphism group of the free group generated by the braid group and the permutation group. This is proved to be the same as the subgroup of automorphisms of permutation-conjugacy type and is represented by generalised braids (braids in which some crossings are allowed to be “welded”). As a consequence of this representation there is a finite presentation which shows the close connection with both the classical braid and permutation groups. The group is isomorphic to the automorphism group of the free quandle and closely related to the automorphism group of the free rack. These automorphism groups are connected with invariants of classical knots and links in the 3-sphere.

Roger Fenn	Related
Richard Rimányi	Related

R. Fenn, M. T. Greene, D. Rolfsen, C. Rourke, and B. Wiest: “Ordering the braid groups,” Pac. J. Math. 191 : 1 (1999), pp. 49–74. MR 1725462 Zbl 1009.20042 article

We give an explicit geometric argument that Artin’s braid group Bn is right-orderable. The construction is elementary, natural, and leads to a new, effectively computable, canonical form for braids which we call left-consistent canonical form. The left-consistent form of a braid which is positive (respectively negative) in our order has consistently positive (respectively negative) exponent in the smallest braid generator which occurs. It follows that our ordering is identical to that of Dehornoy (1995) constructed by very different means, and we recover Dehornoy’s main theorem that any braid can be put into such a form using either positive or negative exponent in the smallest generator but not both.

Our definition of order is strongly connected with Mosher’s (1995) normal form and this leads to an algorithm to decide whether a given braid is positive, trivial, or negative which is quadratic in the length of the braid word.

Dale P. O. Rolfsen	Related
Roger Fenn	Related
Michael Thomas Greene	Related
Bertold Wiest	Related

R. Fenn, C. Rourke, and B. Sanderson: “James bundles,” Proc. Lond. Math. Soc. (3) 89 : 1 (2004), pp. 217–240. MR 2063665 Zbl 1055.55005 article

We study cubical sets without degeneracies, which we call \( \square \)-sets. These sets arise naturally in a number of settings and they have a beautiful intrinsic geometry; in particular a \( \square \)-set \( C \) has an infinite family of associated \( \square \)-sets \( J^i (C), \) for \( i = 1, 2,\dots \), which we call James complexes. There are mock bundle projections \[ p_i: |J^i(C)|\to |C| \] (which we call James bundles) defining classes in unstable cohomotopy which generalise the classical James–Hopf invariants of \( \Omega (S^2) \). The algebra of these classes mimics the algebra of the cohomotopy of \( \Omega (S^2) \) and the reduction to cohomology defines a sequence of natural characteristic classes for a \( \square \)-set. An associated map to \( BO \) leads to a generalised cohomology theory with geometric interpretation similar to that for Mahowald orientation.

Roger Fenn	Related
Brian Joseph Sanderson	Related

R. Fenn, C. Rourke, and B. Sanderson: “The rack space,” Trans. Amer. Math. Soc. 359 : 2 (2007), pp. 701–740. MR 2255194 Zbl 1123.55006 article

The main result of this paper is a new classification theorem for links (smooth embeddings in codimension 2). The classifying space is the rack space and the classifying bundle is the first James bundle. We investigate the algebraic topology of this classifying space and report on calculations given elsewhere. Apart from defining many new knot and link invariants (including generalised James–Hopf invariants), the classification theorem has some unexpected applications. We give a combinatorial interpretation for \( \pi_2 \) of a complex which can be used for calculations and some new interpretations of the higher homotopy groups of the 3-sphere. We also give a cobordism classification of virtual links.

Roger Fenn	Related
Brian Joseph Sanderson	Related

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Colin P. Rourke

Colin Rourke

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