Polynomial values, the linking form and unknotting numbers

We show how the signed evaluations of link polynomials can be used to calculate unknotting numbers. We use the %Lickorish-Millett value of the %Jones polynomial to show that any achiral knot with determinant %divisible by $3$ does not have unknotting number one, and Jones-Rong value of the Brandt-Lickorish-Millett-Ho polynomial $Q$ to calculate the unknotting numbers of $8_{16}$, $9_{49}$ and 6 further new entries in Kawauchi’s tables. Another method is developed by applying and extending the linking form criterion of Lickorish. This leads to several conjectured relations between the Jones-Rong value of $Q$ and the linking form.

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Published 1 January 2004