Twisted Alexander polynomials and surjectivity of a group homomorphism.

Kitano, Teruaki; Suzuki, Masaaki; Wada, Masaaki

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A partial order in the knot table.

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We prove that Wada’s twisted Alexander polynomial of a knot group associated to any nonabelian $S L (2, 𝔽)$ -representation is a polynomial. As a corollary, we show that it is always a monic polynomial of degree $4 g - 2$ for a fibered knot of genus $g$ .

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Positive knots, closed braids and the Jones polynomial

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Using the recent Gauß diagram formulas for Vassiliev invariants of Polyak-Viro-Fiedler and combining these formulas with the Bennequin inequality, we prove several inequalities for positive knots relating their Vassiliev invariants, genus and degrees of the Jones polynomial. As a consequence, we prove that for any of the polynomials of Alexander/Conway, Jones, HOMFLY, Brandt-Lickorish-Millett-Ho and Kauffman there are only finitely many positive knots with the same polynomial and no...