Hi. This is a question about whether it is possible to manually input (or modify) in sage an explicit rewriting system of an already inputed finitely presented group.
Here is the preamble to the more explicit details for my question: I am working in Sage with the fundamental group (call it G) of the double torus. The standard presentation for this group has generators a1, b1, a2, b2, and relator a1 b1 a1^-1 b1^-1 a2 b2 a2^-1 b2^-1. The sage command k = G.rewriting_system() produces the corresponding rewriting system k, which is not confluent (as reported by sage). There is, of course, the sage instruction k.make_confluent() to replace k by a confluent rewriting system, using (I believe) the Knuth-Bendix (KB) algorithm. [Relevant here is the standard fact that the KB-algorithm can be, in general, highly time-consuming, and might actually never end in concrete examples. Further, I believe it is not known whether the above presentation for G will be such a case of a non-finishing procedure with k.make_confluent().] Now, regarding the group G above, it is known that the expanded presentation of G with generators a1, b1, a2, b2, i1, j1, i2, j2 (the last four will represent the inverses of the first four), and relations a1 i1, i1 a1 a2 i2 i2 a2 b1 j1 j1 b1 b2 j2 j2 b2 a1 b1 a1^-1 b1^-1 a2 b2 a2^-1 b2^-1 has a corresponding rewriting system for which the KB algorithm DOES finish producing a well know confluent rewriting system ---described in the PhD thesis (and corresponding paper "Rewriting systems for Coxeter groups") of Susan Hermiller. But a computational problem comes at this point, as the sage instruction k.make_confluent keeps on working for lots of time without producing a final answer in the case of the extended presentation of G above. And yet, as noted above, the expected rewriting system is known. So, my question is: Is there a way to manually input in sage the right (confluent) rewriting system for G? ---instead of wait for apparently an extra long calculation with the instruction make_confluent. An alternative which would work equally well for my needs is the following: Is it possible to input an explicit rewriting system in sage WITHOUT previously inputing a finitely presente group? The idea here would be to directly input the right confluent rewriting system, with which sage produces the corresponding group (G in this case), and then a user could just care about playing with elements in G. By the way, this question is directly related to a current research I am doing about the so-called effective topological complexity of surfaces. This is a concept which rises from an algebraic-topology model for the motion planning problem in robotics. Thanks for any comments and suggestions! -- You received this message because you are subscribed to the Google Groups "sage-support" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/d/optout.
