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On twist positive braids with the $2$ -variable link invariant: $3$ -braids of width $2$

Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Corresponding author.

Department of Mathematical Sciences, College of Science, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia

E-mail Address: sabajri@pnu.edu.sa

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Nouf Abdulrahman Alqahtani

Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn, Saud Islamic University (IMSIU), Saudi Arabia

E-mail Address: naswedan@imamu.edu.sa

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https://doi.org/10.1142/S0218216523500864Cited by:0 (Source: Crossref)

Abstract

The possible $2$ -variable link polynomial $P_{L} (v, z)$ for an oriented link $L$ , which has width $2$ in the variable $v,$ is studied, where width of $P_{L} (v, z)$ is the minimal number of strands allowed by the index bound. It is shown that if the 2-variable link invariant $P_{L} (v, z)$ for an oriented link $L$ has width $k$ in the variable $v,$ then it is the same as the polynomial of a closed $k$ -braid, $k = 1, 2 .$ Also a complete list of $3$ - braids of width $2,$ which close to knots, are given. Consequently it is shown that $P_{L} (v, z)$ determines $\hat{β}$ for the full $3$ -braid $β .$ Finally, the 2-variable polynomial for the non-full 3-braid is calculated.

Keywords:

AMSC: 20F36, 57M05, 57K10

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