Gröbner-Shirshov Bases for the Braid Semigroup

Abstract

We found Gröbner-Shirshov basis for the braid semigroup B + n+1 . It gives a new algorithm for the solution of the word problem for the braid semigroup and so for the braid group.

Describing semigroups with defining relations of the form $$xy=yz$$ x y = y z and $$yx=zy$$ y x = z y and connections with knot theory

“…Whereas knot diagrams are not as ubiquitous as graphs, they have attracted much attention of algebraists in the last century, and knot semigroups described in this paper can become a new natural way of dening semigroups corresponding to knot diagrams; we discuss this further in Section 8. Each relation dening a knot semigroup has words of the same length on the two sides of the equality; such relations are called homogeneous and, accordingly, semigroups dened in this way are also sometimes called homogeneous; another example of homogeneous semigroups are braid semigroups (for their denition see, for example, [2]); for a brief review of more examples of classes of homogeneous semigroups see [3]. There has been a number of attempts to dene conjugate elements in semigroups, generalising conjugation in groups.…”

“…With the exception of some trivial cases, knot monoids are neither Garside monoids nor divisibility monoids. However, all knot monoids which we describe in this paper are thin 2 . The aim of this paper is to consider several standard types of knot diagrams and explicitly describe their knot semigroups (by the way, our description solves the word problem in the semigroups).…”

“…In this section we describe the knot semigroup Kt n of a knot diagram t n (with an odd n); the knot semigroup of a link diagram t n (with an even n) has a slightly more complicated structure, which we describe in Section 6. It is worth mentioning briey that a similar diagram consisting of n clockwise half-twists can be considered; it is known as a right-handed torus knot, whereas a 5 Note that the term`braid semigroup' already has a well established meaning, namely, it is the subsemigroup of the braid group generated by clockwise half-twists (see, for example, [2]). …”

Hilbert Series of Positive Braids

“…In 2003, Bokut et al [5] gave a non-commutative Gröbner basis or complete presentation of the braid monoid MB n+1 and proved (with the length-lexicographic order induced by x 1 < x 2 < · · · < x n ):…”

Gröbner–Shirshov basis for the Braid group in the Birman–Ko–Lee generators

“…In the paper [21], a Gröbner-Shirshov basis of the semigroup of positive braids B + n in the Artin generators was found. Based on this paper, in the paper [24], I found a Gröbner-Shirshov basis of the braid group B n+1 in the Artin-Garside generators a i , 1 i n, Δ, Δ −1 , where [36]).…”