Math. Z. 224, 255 – 287 (1997) A new characterisation of Braid groups and rings of invariants for symmetric automorphism groups Stephen P. Humphries Department of Mathematics, Brigham Young University, Provo, Utah, 84602, USA Received 17 May 1994; in nal form 7 April 1995 1 Introduction Let F (n)= 〈x 1 ;:::;x n 〉 be a free group of rank n with xed free generating set x 1 ;:::;x n and Aut(F (n)) its automorphism group. An endomorphism  of F (n) is symmetric (relative to this choice of x 1 ;:::;x n ) if (x i )= w i x j w −1 i for i =1;:::;n, where i → j is an endomorphism of {1;:::;n} and w i ∈ F (n) for i =1;:::;n. The monoid of all symmetric endomorphisms of F (n) is denoted by SE(n). There is an epimorphism of monoids SE(n) → End(n), where End(n) is the monoid of endomorphisms of {1;:::;n}, whose kernel is PSE(n) the pure symmetric endomorphisms of F (n). The group of symmetric automorphisms is ˆ H (n) = Aut(F (n)) ∩ SE(n) and we let H(n) = Aut(F (n)) ∩ PSE(n). Then ˆ H (n) contains the braid groups B n [Bi1, p.30] and H (n) contains the pure braid group P n = B n ∩ PSE(n). The braid groups give an algebraic way of looking at knots and links [Bi1], while the groups ˆ H (n) are related to singular knots and links where certain strands are ‘welded’ [F-R-R, Bi2]. In [Hu3] we described the action of ˆ H (n), and so of B n , as automorphisms of a nitely generated polynomial algebra. In this paper we give conditions which guarantee that an element of ˆ H (n), thought of as acting on this polyno- mial algebra, is in the image of B n . We then use this characterisation to study geometric questions concerning closed braids, in particular Markov’s second move. We also determine the ring of invariants for this (non-linear) action of ˆ H (n) on the polynomial algebra and describe interesting elements of this algebra which are invariant under the action of B n . The groups H (n) are generated [Hu1] by elements t ij : t ij (x k )= x k if k  j; t ij (x j )= x i x j x −1 i : A presentation for H (n) (with these generators) is given in the paper of McCool [Mc]; but see also [F-R]. Generators for ˆ H (n) are obtained by also including permutation elements (of the symmetric group S n ) acting naturally on