Three-page approach to knot theory. Universal semigroup
In a previous paper [Funct. Anal. Appl. 33, No. 4, 260-269 (1999); translation from Funkts. Anal. Prilozh. 33, No. 4, 25-37 (1999; Zbl 0947.57005)], the author discussed three-page link diagrams as links embedded in a three-page book and a new way to encode nonoriented links in the three-page book by words in a \(12\)-letter alphabet and presented a finite set of local moves realizing the equivalence of link types. In the paper under review, the author generalizes the notion of three-page link diagram. More precisely, the author introduces the three-page tangle diagrams which form a finitely presented semigroup whose center classifies the nonoriented links in the three-space. The author also gives a finite presentation for the group of invertible elements of the semigroup and shows that the commutator subgroup of this group is isomorphic to the braid group of infinitely many strands. Finally, the Kauffman theorem on the existence and uniqueness of the \(L\)-polynomial invariant is stated in the three-page language.
scientific article; zbMATH DE number 1497238