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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Jones polynomials of alternating links


Author: Kunio Murasugi
Journal: Trans. Amer. Math. Soc. 295 (1986), 147-174
MSC: Primary 57M25
MathSciNet review: 831194
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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

DOI: http://dx.doi.org/10.1090/S0002-9947-1986-0831194-9
PII: S 0002-9947(1986)0831194-9
Keywords: Knot, link, Jones polynomial, alternating knot, reduced Alexander polynomial
Article copyright: © Copyright 1986 American Mathematical Society