A new obstruction of quasi-alternating links

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We prove that the degree of the Brandt-Lickorish-Millet polynomial of any quasi-alternating link is less than its determinant. Therefore, we obtain a new and a simple obstruction criterion for quasi-alternateness. As an application, we identify some knots of 12 crossings or less and some links of 9 crossings or less that are not quasi-alternating. Also, we show that there are only finitely many Kanenobu knots which are quasi-alternating. This last result supports Conjecture 3.1 of Greene in [10] which states that there are only finitely many quasi-alternating links with a given determinant. Moreover, we identify an infinite family of non quasi-alternating Montesinos links and this supports Conjecture 3.10 in [20] that characterizes quasi-alternating Montesinos links.