Racks and Links in Codimension Two

Abstract

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.

Twisted quandle homology theory and cocycle knot invariants

“…This definition of colorings on knot diagrams has been known, see [13,17] for example. Henceforth, all the quandles that are used to color diagrams will be finite.…”

Quandle and Biquandle Homology Calculation in R

“…In [2], David Joyce showed that racks are indeed a knot invariant. Subsequently, Fenn and Rourke introduced racks as proper knot invariants in [3]. It is instructive to consider racks as groups without their multiplicative elements.…”

Triple point numbers of surface-links and symmetric quandle cocycle invariants

“…A quandle (see Fenn and Rourke [3], Joyce [5] or Matveev [10]) is a set X with a binary operation .x; y/ 7 ! x y such that (i) for any x 2 X , it holds that x x D x , (ii) for any x; y 2 X , there exists a unique z 2 X such that z y D x , and (iii) for any x; y; z 2 X , it holds that .x y / z D .x z / .y z / .…”