Regular ArticleNo Access

The degree 2 part of the LMO invariant of cyclic branched covers 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/S0129167X24500125Cited by:0 (Source: Crossref)

Abstract

The LMO invariant is a universal quantum invariant of closed oriented 3-manifolds. In this paper, we present the degree 2 part of the LMO invariant of cyclic branched covers of knots by using the 3-loop invariant of knots, and we calculate it concretely for knots obtained by plumbing the doubles of two knots.

Communicated by Yasuyuki Kawahigashi

Keywords:

AMSC: 57K31, 57K10

References

1. D. Bar-Natan and R. Lawrence , A rational surgery formula for the LMO invariant, Israel J. Math. 140 (2004) 29–60. Crossref, Web of Science, Google Scholar
2. D. Bar-Natan, T. T. Q. Le and D. P. Thurston , Two applications of elementary knot theory to Lie algebras and Vassiliev invariants, Geom, Topol. 7 (2003) 1–31. Crossref, Web of Science, Google Scholar
3. S. Garoufalidis and A. Kricker , A rational noncommutative invariant of boundary links, Algebra Geom. Topol. 8 (2004) 115–204. Crossref, Web of Science, Google Scholar
4. S. Garoufalidis and A. Kricker , Finite type invariants of cyclic branched covers, Topology 43 (2004) 1247–1283. Crossref, Google Scholar
5. T. Ito , On LMO invariant constraints for cosmetic surgery and other surgery problems for knots in $S^{3}$ , Comm. Anal. Geom. 28(2) (2020) 321–349. Crossref, Web of Science, Google Scholar
6. L. Kauffman , On knots, Ann. Math. Stud. 115 (1987), https://press.princeton.edu/books/paperback/9780691084350/on-knots-am-115-volume-115. Google Scholar
7. A. Kricker, The lines of the Kontsevich integral and Rozansky’s rationality conjecture, preprint (2000), arXiv:math. GT/0005284. Google Scholar
8. A. Kricker , A surgery formula for the 2-loop piece of the LMO invariant of a pair, in Invariants of Knots and 3-Manifolds (Kyoto 2001), Geometry and Topology Monographs, Vol. 4 (Geomentry Topology Publication, Coventry, 2002), pp. 161–181. Crossref, Google Scholar
9. T. T. Q. Le, J. Murakami and T. Ohtsuki , A universal quantum invariant of 3-manifolds, Topology 37 (1998) 539–574. Crossref, Web of Science, Google Scholar
10. T. Ohtsuki , Quantum Invariants, Series on Knots and Everything, Vol. 29 (World Scientific Publishing, River Edge, New Jersey, 2002). Google Scholar
11. T. Ohtsuki and D. Moskovich , Vanishing of 3-loop Jacobi diagrams of odd degree, J. Combin. Theory Ser. A 114 (2007) 919–930. Crossref, Web of Science, Google Scholar
12. C. Sato , Casson-Walker invariant of Seifert fibered rational homology spheres as quantum $S O (3)$ -invariant, J. Knot Theory Ramifications 6(1) (1997) 79–93. Link, Web of Science, Google Scholar
13. K. Yamaguchi , The 3-loop polynomial of knots obtained by plumbing the doubles of two knots, Int. J. Math. 34(14) (2023) 2350086. Link, Web of Science, Google Scholar