Pseudocharacters of braid groups and prime links

Abstract

Abstract. Pseudocharacters of groups have recently found an application in the theory of classical knots and links in R 3 . More precisely, there is a relationship between pseudocharacters of Artin's braid groups and the properties of links represented by braids. In the paper, this relationship is investigated and the notion of kernel pseudocharacters of braid groups is introduced. It is proved that if a kernel pseudocharacter φ and a braid β satisfy |φ(β)| > C φ , where C φ is the defect of φ, then β represents a prime link (i.e., a link that is noncomposite, nonsplit, and nontrivial). Furthermore, the space of braid group pseudocharacters is studied and a way is described to obtain nontrivial kernel pseudocharacters from an arbitrary braid group pseudocharacter that is not a homomorphism. This makes it possible to employ an arbitrary nontrivial braid group pseudocharacter for the recognition of prime knots and links.

On independence of some pseudocharacters on braid groups

“…Theorem 2 (see [2]). Let H be a normal subgroup of finite index in a group G, and let χ be a pseudocharacter on H such that (2) χ(ghg −1 ) = χ(h) for all g ∈ G, h ∈ H. Then χ extends uniquely to a pseudocharacter on the entire group G.…”

“…In the present paper we shall consider the Artin braid group B n on n strands; the above theorem will be applied to the case where H = P n , the pure braid subgroup. Thus, if some pseudocharacter is already defined for pure braids on n strands and condition (2) is satisfied, then we tacitly assume it to be extended to the entire braid group.…”

“…Since del i is a homomorphism of P n to P n−1 , the operator R takes pseudocharacters to pseudocharacters. Summation over all strands ensures condition (2). Then the pseudocharacter Rχ extends to the entire group B n in the standard way.…”

“…In [2], Malyutin showed how kernel pseudocharacters can be used for proving that some Markov type operations cannot be applied to a braid.…”

Coloring link diagrams and Conway-type polynomial of braids

“…Recently, invariants of conjugacy classes of braids received a considerable attention, since in some cases they define quasi-morphisms on braid groups and induce quasi-morphisms on certain groups of diffeomorphisms of smooth manifolds, see for example [3,6,7,8,11,12,14,15,19,20].…”