Twisted alexander polynomials on curves in character varieties of knot groups

Taehee Kim, Takahiro Kitayama, Takayuki Morifuji

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

For a fibered knot in the 3-sphere the twisted Alexander polynomial associated to an SL(2)-character is known to be monic. It is conjectured that for a nonfibered knot there is a curve component of the SL(2)-character variety containing only finitely many characters whose twisted Alexander polynomials are monic, i.e. finiteness of such characters detects fiberedness of knots. In this paper, we discuss the existence of a certain curve component which relates to the conjecture when knots have nonmonic Alexander polynomials. We also discuss the similar problem of detecting the knot genus.

Original languageEnglish
Article number1350022
JournalInternational Journal of Mathematics
Volume24
Issue number3
DOIs
Publication statusPublished - 2013 Mar

Fingerprint

Character Variety
Knot Group
Alexander Polynomial
Knot
Monic
Curve
Fibered Knot
Finiteness
Genus
Character

Keywords

  • 57M05
  • 57M25
  • character variety
  • fibered knot 57M27
  • Twisted Alexander polynomial

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Twisted alexander polynomials on curves in character varieties of knot groups. / Kim, Taehee; Kitayama, Takahiro; Morifuji, Takayuki.

In: International Journal of Mathematics, Vol. 24, No. 3, 1350022, 03.2013.

Research output: Contribution to journalArticle

@article{e84eb6c4fb0c443a9406112e88fccb6c,
title = "Twisted alexander polynomials on curves in character varieties of knot groups",
abstract = "For a fibered knot in the 3-sphere the twisted Alexander polynomial associated to an SL(2)-character is known to be monic. It is conjectured that for a nonfibered knot there is a curve component of the SL(2)-character variety containing only finitely many characters whose twisted Alexander polynomials are monic, i.e. finiteness of such characters detects fiberedness of knots. In this paper, we discuss the existence of a certain curve component which relates to the conjecture when knots have nonmonic Alexander polynomials. We also discuss the similar problem of detecting the knot genus.",
keywords = "57M05, 57M25, character variety, fibered knot 57M27, Twisted Alexander polynomial",
author = "Taehee Kim and Takahiro Kitayama and Takayuki Morifuji",
year = "2013",
month = "3",
doi = "10.1142/S0129167X13500225",
language = "English",
volume = "24",
journal = "International Journal of Mathematics",
issn = "0129-167X",
publisher = "World Scientific Publishing Co. Pte Ltd",
number = "3",

}

TY - JOUR

T1 - Twisted alexander polynomials on curves in character varieties of knot groups

AU - Kim, Taehee

AU - Kitayama, Takahiro

AU - Morifuji, Takayuki

PY - 2013/3

Y1 - 2013/3

N2 - For a fibered knot in the 3-sphere the twisted Alexander polynomial associated to an SL(2)-character is known to be monic. It is conjectured that for a nonfibered knot there is a curve component of the SL(2)-character variety containing only finitely many characters whose twisted Alexander polynomials are monic, i.e. finiteness of such characters detects fiberedness of knots. In this paper, we discuss the existence of a certain curve component which relates to the conjecture when knots have nonmonic Alexander polynomials. We also discuss the similar problem of detecting the knot genus.

AB - For a fibered knot in the 3-sphere the twisted Alexander polynomial associated to an SL(2)-character is known to be monic. It is conjectured that for a nonfibered knot there is a curve component of the SL(2)-character variety containing only finitely many characters whose twisted Alexander polynomials are monic, i.e. finiteness of such characters detects fiberedness of knots. In this paper, we discuss the existence of a certain curve component which relates to the conjecture when knots have nonmonic Alexander polynomials. We also discuss the similar problem of detecting the knot genus.

KW - 57M05

KW - 57M25

KW - character variety

KW - fibered knot 57M27

KW - Twisted Alexander polynomial

UR - http://www.scopus.com/inward/record.url?scp=84876733026&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84876733026&partnerID=8YFLogxK

U2 - 10.1142/S0129167X13500225

DO - 10.1142/S0129167X13500225

M3 - Article

AN - SCOPUS:84876733026

VL - 24

JO - International Journal of Mathematics

JF - International Journal of Mathematics

SN - 0129-167X

IS - 3

M1 - 1350022

ER -