On minimal crossing number braid diagrams and homogeneous braids

Ilya Alekseev, Geidar Mamedov

Research output


We study braid diagrams with a minimal number of crossings. Such braid diagrams correspond to geodesic words for braid groups Bn with standard Artin generators. We present a braid version of the celebrated Tait conjectures on alternating links. More precisely, we prove that a diagram of an alternating braid is minimal if and only if it is alternating and that any two minimal diagrams of the same alternating braid are isotopic. Using this, we prove that monoids of alternating braids are right–angled Artin monoids. We conjecture that similar results hold true for a more general class of homogeneous braids. In particular, we prove that any homogeneous braid diagram is minimal. Also, we study the growth and geodesic growth functions of Bn. Finally, we give sufficient
conditions on a braid diagram to be minimal and discuss some relations between our results and an open problem in combinatorial group theory of the braid groups, stated in Kirby’s List [1].
Original languageEnglish
Article number03210
Number of pages14
Publication statusE-pub ahead of print - 8 May 2019

Scopus subject areas

  • Algebra and Number Theory

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