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We show that a Brunnian link of      n n       components and the      n n       component trivial link share the same first      n −  1 n-1       coefficients of the Jones-Conway (Homflypt) polynomial (answering the question of Kanenobu and Miyazawa). We prove also the similar result for the Kauffman polynomial of Brunnian links. We place our solution in the context of Vassiliev-Gusarov skein modules based on mixed singular crossings.

The Kanenobu-Miyazawa conjecture and the Vassiliev-Gusarov skein modules based on mixed crossings