From colored Jones invariants to logarithmic invariants

Jun Murakami

doi:10.3836/tjm/1502179244

From colored Jones invariants to logarithmic invariants

Jun Murakami^*

^*Corresponding author for this work

School of Fundamental Science and Engineering

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

In this paper, we express the logarithmic invariant of knots in terms of derivatives of the colored Jones invariants. Logarithmic invariant is defined by using the Jacobson radicals of the restricted quantum group Ū_ξ (sl₂) where ξ is a root of unity. We also propose a version of the volume conjecture stating a relation between the logarithmic invariants and the hyperbolic volumes of the cone manifolds along a knot, which is proved for the figure-eight knot.

Original language	English
Pages (from-to)	453-475
Number of pages	23
Journal	Tokyo Journal of Mathematics
Volume	41
Issue number	2
DOIs	https://doi.org/10.3836/tjm/1502179244
Publication status	Published - 2018

Keywords

Hyperbolic volume
Knot theory
Quantum group

ASJC Scopus subject areas

Mathematics(all)

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10.3836/tjm/1502179244

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title = "From colored Jones invariants to logarithmic invariants",

abstract = "In this paper, we express the logarithmic invariant of knots in terms of derivatives of the colored Jones invariants. Logarithmic invariant is defined by using the Jacobson radicals of the restricted quantum group Ūξ (sl2) where ξ is a root of unity. We also propose a version of the volume conjecture stating a relation between the logarithmic invariants and the hyperbolic volumes of the cone manifolds along a knot, which is proved for the figure-eight knot.",

keywords = "Hyperbolic volume, Knot theory, Quantum group",

author = "Jun Murakami",

note = "Funding Information: I would like to thank Gregor Masbaum for valuable discussion. This research is partially supported by the Grant-in-Aid for Scientific Research (B) (25287014), Exploratory Research (25610022) of Japan Society for the Promotion of Science, and the Erwin Schr{\"o}dinger Institute for Mathematical Physics (ESI) in Vienna. Publisher Copyright: {\textcopyright} 2018 International Academic Printing Co. Ltd.. All rights reserved.",

year = "2018",

doi = "10.3836/tjm/1502179244",

language = "English",

volume = "41",

pages = "453--475",

journal = "Tokyo Journal of Mathematics",

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T1 - From colored Jones invariants to logarithmic invariants

AU - Murakami, Jun

N1 - Funding Information: I would like to thank Gregor Masbaum for valuable discussion. This research is partially supported by the Grant-in-Aid for Scientific Research (B) (25287014), Exploratory Research (25610022) of Japan Society for the Promotion of Science, and the Erwin Schrödinger Institute for Mathematical Physics (ESI) in Vienna. Publisher Copyright: © 2018 International Academic Printing Co. Ltd.. All rights reserved.

PY - 2018

Y1 - 2018

N2 - In this paper, we express the logarithmic invariant of knots in terms of derivatives of the colored Jones invariants. Logarithmic invariant is defined by using the Jacobson radicals of the restricted quantum group Ūξ (sl2) where ξ is a root of unity. We also propose a version of the volume conjecture stating a relation between the logarithmic invariants and the hyperbolic volumes of the cone manifolds along a knot, which is proved for the figure-eight knot.

AB - In this paper, we express the logarithmic invariant of knots in terms of derivatives of the colored Jones invariants. Logarithmic invariant is defined by using the Jacobson radicals of the restricted quantum group Ūξ (sl2) where ξ is a root of unity. We also propose a version of the volume conjecture stating a relation between the logarithmic invariants and the hyperbolic volumes of the cone manifolds along a knot, which is proved for the figure-eight knot.

KW - Hyperbolic volume

KW - Knot theory

KW - Quantum group

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DO - 10.3836/tjm/1502179244

M3 - Article

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SN - 0387-3870

VL - 41

SP - 453

EP - 475

JO - Tokyo Journal of Mathematics

JF - Tokyo Journal of Mathematics

IS - 2

ER -

From colored Jones invariants to logarithmic invariants

Abstract

Keywords

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