Abstract
It is well-known that the second highest coefficient of the Alexander polynomial of any lens space knot in $S^3$ is −1. We show that if the third highest coefficient of the Alexander polynomial $\Delta_K(t)$ of a lens space knot $K$ in $S^3$ is non-zero, then $\Delta_K(t)$ coincides with the Alexander polynomial of the $(2,2g+1)$-torus knot, where $g$ is the Seifert genus of $K$.
Funding Statement
The author is partially supported by Grant-in-aid for Science
Research, No. 17K14180.
Acknowledgement
Question 1 was presented at Masakazu Teragaito’s talk in the Minisymposium ‘‘Knot Theory on Okinawa’’ at OIST from February 17–21, 2020. I thank him for telling me this question. I am grateful to anonymous referees for giving several useful comments and suggestions for my first manuscript.
Citation
Motoo Tange. "The third term in lens surgery polynomials." Hiroshima Math. J. 51 (1) 101 - 109, March 2021. https://doi.org/10.32917/h2020050
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