caa a22 4500 475740750 CHVBK 20180406123459.0 cr unu---uuuuu 170329e20001101xx s 000 0 eng 10.1023/A:1026667407199 doi (NATIONALLICENCE)springer-10.1023/A:1026667407199 Orevkov S. V. A. Steklov Mathematics Institute, Russian Academy of Sciences, Russia aut Strong Positivity in Right-Invariant Order on Braid Groups and Quasipositivity [Elektronische Daten] [S. Orevkov] Dehornoy constructed a right invariant order on the braid group B n uniquely defined by the condition $$\beta _0 \sigma _i \beta _1 >1{\text{ if }}\beta _0 ,\beta _1$$ are words in $$\sigma _{i + 1}^{ \pm 1} ,...,\sigma _{n - 1}^{ \pm 1}$$ . A braid is called strongly positive if $$\alpha \beta \alpha ^{ - 1} >1$$ for any $$\alpha \in B_n$$ . In the present paper it is proved that the braid $$\beta _0 \left( {\sigma _1 \sigma _2 ...\sigma _{n - 1} } \right)\left( {\sigma _{n - 1} \sigma _{n - 2} ...\sigma _1 } \right)$$ is strongly positive if the word $$\beta _0$$ does not contain $$\sigma _1^{ \pm 1}$$ . We also provide a geometric proof of the result by Burckel and Laver that the standard generators of a braid group are strongly positive. Finally, we discuss relations between the right invariant order and quasipositivity. Plenum Publishing Corporation, 2000 braid nationallicence right-invariant order nationallicence quasipositive braid nationallicence curve diagram nationallicence Mathematical Notes Kluwer Academic Publishers-Plenum Publishers 68/5-6(2000-11-01), 588-593 0001-4346 68:5-6<588 2000 68 11006 https://doi.org/10.1023/A:1026667407199 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1023/A:1026667407199 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Orevkov S. V. A. Steklov Mathematics Institute, Russian Academy of Sciences, Russia aut NATIONALLICENCE 773 0- Mathematical Notes Kluwer Academic Publishers-Plenum Publishers 68/5-6(2000-11-01), 588-593 0001-4346 68:5-6<588 2000 68 11006 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer