Finitely presented groups and semigroups in knot theory (Q2783054)

The author obtains an algebraic classification of the isotopy classes of non-oriented links. With this goal, he considers tangles embedded in a book with countably many pages and defines a semigroup \(Y_\infty\) by explicitly giving generators and relations (\(Y_\infty\) is not-finitely presented). The main theorem states that each embedded tangle, up to isotopy, can be represented by a word in the generators of \(Y_\infty\); moreover, if two words correspond to isotopic tangles, then they coincide in \(Y_\infty\). Analogously, the embedding of tangles in books with finitely many pages allows to define finitely presented subsemigroups of \(Y_\infty\), denoted by \(Y_n\) (\(n\geq 3\)), with the following properties: (a) the center of \(Y_n\) is isomorphic to the semigroup of the isotopy classes of links; (b) the group of the invertible elements of \(Y_n\) contains a subgroup which is isomorphic to the braid group on infinitely many strands. As a consequence, an algebraic proof of the solvability of the word problem for \(Y_n\) allows to obtain an algorithm for recognizing links, and vice-versa.NEWLINENEWLINEFor the entire collection see [Zbl 0981.00006].