Homological Characterization of the Unknot

Given a knot K in the 3-sphere, let QK be its fundamental quandle as introduced by D. Joyce. Its first homology group is easily seen to be H1(QK) ∼= Z. We prove that H2(QK) = 0 if and only if K is trivial, and H2(QK) ∼= Z whenever K is non-trivial. An analogous result holds for links, thus characterizing trivial components. More detailed information can be derived from the conjugation quandle: let QK be the conjugacy class of a meridian in the knot group π1(SrK). We show that H2(QK) ∼= Zp, where p is the number of prime summands in a connected sum decomposition of K. Introduction and statement of results The fundamental group of a knot. For a knot K in the 3-sphere S let πK := π1(SrK) be the fundamental group of the knot complement. All higher homotopy groups vanish [27], which means that SrK is an Eilenberg-MacLane space. By Poincaré duality, its integral homology is given by H0 ∼= H1 ∼= Z and Hn = 0 for all n ≥ 2. This means that among these classical invariants of algebraic topology, only the group πK contains information about the knot K. The knot group is indeed a very strong invariant: it classifies unoriented prime knots [15, 33]. To capture the complete information, one can consider a meridianlongitude pairmK , lK ∈ πK (see §1). It follows from the work of F.Waldhausen [31] that the group system (πK ,mK , lK) classifies knots. The fundamental quandle of a knot. A quandle, as introduced by D. Joyce [19], is a set Q with a binary operation whose axioms model conjugation in a group, or equivalently, the Reidemeister moves of knot diagrams. Quandles have been intensively studied by different authors and under various names: as “distributive groupoids” by S.V.Matveev [22], as “crossed G-sets” by P.J. Freyd and D.N.Yetter [14], as “crystals” by L.H.Kauffman [20], and — slightly generalized — as “automorphic sets” by E.Brieskorn [1], and as “racks” by R. Fenn and C.Rourke [12]. We review the relevant definitions in §2. The Wirtinger presentation of the knot group πK involves only conjugation and thus may be re-interpreted as defining a quandle. The quandle QK so presented is called the fundamental quandle of the knot K (see §2). Using Waldhausen’s results, Joyce [19] showed that the knot quandle is a classifying invariant: if QK and QK′ are isomorphic, then the knots K and K ′ are equivalent up to inversion. Date: June 30, 2002 (revised version). 1991 Mathematics Subject Classification. 57M25, 57M05, 55N99.