A New Way to Represent Links. One-Dimensional Formalism and Untangling Technology

An alternative link representation different from planar diagrams is discussed. Isotopy classes of unordered nonoriented links are realized as central elements of a monoid presented explicitly by a finite number of generators and relations. The group presented by two generators and three relations a,b],a ² ba ^–2]=a,b],b ² ab ^–2]=a,b],a ^–1,b ^–1]]=1, where x,y]=xyx ^–1 y ^–1, is proved to have a commutator subgroup isomorphic to the braid group on infinitely many strands. A new partial algorithm for unknot recognition is constructed. Experiments show that the algorithm allows the untangling of unknots whose planar diagram has hundreds of crossings. Here 'untangling' means 'finding an isotopy to the circle'.