TAILIEUCHUNG - Solution of the word problem in the singular braid group

Singular braids are isotopy classes of smooth strings which are allowed to cross each other pairwise with distinct tangents. Under the usual multiplication of braids, they form a monoid. The singular braid group was introduced by FennKeyman-Rourke as the quotient group of the singular braid monoid. We give a solution of the word problem for this group. | Turk J Math 28 (2004) , 95 – 100. ¨ ITAK ˙ c TUB Solution of the Word Problem in the Singular Braid Group Stepan Yu. Orevkov Abstract Singular braids are isotopy classes of smooth strings which are allowed to cross each other pairwise with distinct tangents. Under the usual multiplication of braids, they form a monoid. The singular braid group was introduced by FennKeyman-Rourke as the quotient group of the singular braid monoid. We give a solution of the word problem for this group. It is obtained as a combination of the results by Fenn-Keyman-Rourke and some simple geometric considerations based on the mapping class interpretation of braids. Combined with Corran’s normal form for the singular braid monoid, our algorithm provides a computable normal form for the singular braid group. 1. Introduction Let X be any set. Let us denote X × {1, . . . , n − 1} by Xn . We shall denote an element (x, i) of Xn by xi . Let Σn = {σ1 , . . . , σn−1}. The singular braid group Bn (X)G is the group generated by Xn ∪ Σn and subject to the relations σi σj = σj σi , σi xj = xj σi , σi σj σi = σj σi σj , xi yj = yj xi , xi σ j σ i = σj σ i xj , σi xi = xi σi

• Toán học
• Singular braid group
• Isotopy classes
• Distinct tangents
• Singular braid monoid
• Algorithm provides
• Fuzzy Situation
• Electronic Interaction
• Architecture Framework
• Empirical Investigation
• Commerce Websites
• Filtering Algorithm