The second coefficient of the Alexander polynomial as a satellite obstruction

A set <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝒫</mml:mi> </mml:math> of links is introduced, containing positive braid links as well as arborescent positive Hopf plumbings. It is shown that for links in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝒫</mml:mi> </mml:math> , the leading and the second coefficient of the Alexander polynomial have opposite sign. It follows that certain satellite links, such as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -cables, are not in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝒫</mml:mi> </mml:math> .