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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Jones polynomials of alternating links
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by Kunio Murasugi PDF
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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Additional Information
  • © Copyright 1986 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 295 (1986), 147-174
  • MSC: Primary 57M25
  • DOI: https://doi.org/10.1090/S0002-9947-1986-0831194-9
  • MathSciNet review: 831194