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The 3-loop polynomial of knots obtained by plumbing the doubles of two knots

Research Institute for Mathematical Science, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan

E-mail Address: yamakou@kurims.kyoto-u.ac.jp

https://doi.org/10.1142/S0129167X23500866Cited by:3 (Source: Crossref)

Abstract

The 3-loop invariant (or, the 3-loop polynomial) of a knot is a rational form (or, a polynomial) presenting the 3-loop part of the Kontsevich invariant of knots. In this paper, we calculate the 3-loop polynomial of knots obtained by plumbing the doubles of two knots; this class of knots includes untwisted Whitehead doubles. We construct the 3-loop invariant by calculating the rational version of the Aarhus integral of a surgery presentation. As a consequence, we obtain an explicit presentation of the 3-loop polynomial for the knots.

Communicated by Yasuyuki Kawahigashi

Keywords:

AMSC: 57K10, 57K14, 57K16

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