Jones polynomials of alternating links

Murasugi, Kunio

doi:10.1090/S0002-9947-1986-0831194-9

Jones polynomials of alternating links
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Trans. Amer. Math. Soc. 295 (1986), 147-174 Request permission

Abstract:

Let ${J_K}(t) = {a_r}{t^r} + \cdots + {a_s}{t^s},r > s$, be the Jones polynomial of a knot $K$ in ${S^3}$. For an alternating knot, it is proved that $r - s$ is bounded by the number of double points in any alternating projection of $K$. This upper bound is attained by many alternating knots, including $2$-bridge knots, and therefore, for these knots, $r - s$ gives the minimum number of double points among all alternating projections of $K$. If $K$ is a special alternating knot, it is also proved that ${a_s} = 1$ and $s$ is equal to the genus of $K$. Similar results hold for links.

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Jones polynomials and classical conjectures in knot theory