Algebraic & Geometric Topology

Closed surfaces and character varieties

Eric Chesebro

Full-text: Open access

Abstract

The powerful character variety techniques of Culler and Shalen can be used to find essential surfaces in knot manifolds. We show that module structures on the coordinate ring of the character variety can be used to identify detected boundary slopes as well as when closed surfaces are detected. This approach also yields new number theoretic invariants for the character varieties of knot manifolds.

Article information

Source
Algebr. Geom. Topol., Volume 13, Number 4 (2013), 2001-2037.

Dates
Received: 22 February 2013
Accepted: 3 March 2013
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715628

Digital Object Identifier
doi:10.2140/agt.2013.13.2001

Mathematical Reviews number (MathSciNet)
MR3073906

Zentralblatt MATH identifier
1270.57039

Subjects
Primary: 57M27: Invariants of knots and 3-manifolds

Keywords
3–manifold character variety essential surface

Citation

Chesebro, Eric. Closed surfaces and character varieties. Algebr. Geom. Topol. 13 (2013), no. 4, 2001--2037. doi:10.2140/agt.2013.13.2001. https://projecteuclid.org/euclid.agt/1513715628


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References

  • M F Atiyah, I G Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, MA (1969)
  • S Boyer, X Zhang, On Culler–Shalen seminorms and Dehn filling, Ann. of Math. 148 (1998) 737–801
  • S Boyer, X Zhang, A proof of the finite filling conjecture, J. Differential Geom. 59 (2001) 87–176
  • P J Callahan, M V Hildebrand, J R Weeks, A census of cusped hyperbolic $3$-manifolds, Math. Comp. 68 (1999) 321–332
  • E Chesebro, S Tillmann, Not all boundary slopes are strongly detected by the character variety, Comm. Anal. Geom. 15 (2007) 695–723
  • D Cooper, M Culler, H Gillet, D D Long, P B Shalen, Plane curves associated to character varieties of $3$-manifolds, Invent. Math. 118 (1994) 47–84
  • D Cooper, D D Long, The $A$-polynomial has ones in the corners, Bull. London Math. Soc. 29 (1997) 231–238
  • D Cooper, D D Long, Representation theory and the $A$-polynomial of a knot, Chaos Solitons Fractals 9 (1998) 749–763
  • M Culler, C M Gordon, J Luecke, P B Shalen, Dehn surgery on knots, Ann. of Math. 125 (1987) 237–300
  • M Culler, P B Shalen, Varieties of group representations and splittings of $3$-manifolds, Ann. of Math. 117 (1983) 109–146
  • N M Dunfield, Examples of non-trivial roots of unity at ideal points of hyperbolic $3$-manifolds, Topology 38 (1999) 457–465
  • F González-Acuña, J M Montesinos-Amilibia, On the character variety of group representations in ${\rm SL}(2,{\bf C})$ and ${\rm PSL}(2,{\bf C})$, Math. Z. 214 (1993) 627–652
  • O Goodman, D Heard, C Hodgson, Commensurators of cusped hyperbolic manifolds, Experiment. Math. 17 (2008) 283–306
  • R Hartshorne, Algebraic geometry, Graduate Texts in Mathematics 52, Springer, New York (1977)
  • A Hatcher, On the boundary curves of incompressible surfaces, Pacific J. Math. 99 (1982) 373–377
  • A Hatcher, W P Thurston, Incompressible surfaces in $2$-bridge knot complements, Invent. Math. 79 (1985) 225–246
  • P B Kronheimer, T S Mrowka, Dehn surgery, the fundamental group and SU$(2)$, Math. Res. Lett. 11 (2004) 741–754
  • S Lang, Introduction to algebraic geometry, Addison-Wesley Publishing Co., Reading, MA (1972)
  • C Maclachlan, A W Reid, The arithmetic of hyperbolic 3-manifolds, Graduate Texts in Mathematics 219, Springer, New York (2003)
  • D Mumford, The red book of varieties and schemes, Lecture Notes in Mathematics 1358, Springer, Berlin (1988)
  • U Oertel, Closed incompressible surfaces in complements of star links, Pacific J. Math. 111 (1984) 209–230
  • S Schanuel, X Zhang, Detection of essential surfaces in 3-manifolds with ${\rm SL}\sb 2$-trees, Math. Ann. 320 (2001) 149–165
  • I R Shafarevich, Basic algebraic geometry, I: varieties in projective space, 2nd edition, Springer, Berlin (1994)
  • P B Shalen, Three-manifold topology and the tree for $\rm PSL\sb 2$: the Smith conjecture and beyond, from: “Algebra, $K$-theory, groups, and education”, (T Y Lam, A R Magid, editors), Contemp. Math. 243, Amer. Math. Soc., Providence, RI (1999) 189–209
  • P B Shalen, Representations of 3-manifold groups, from: “Handbook of geometric topology”, (R J Daverman, R B Sher, editors), North-Holland, Amsterdam (2002) 955–1044
  • W P Thurston, Three-dimensional geometry and topology, Vol. 1, Princeton Mathematical Series 35, Princeton Univ. Press (1997)
  • S Tillmann, On the Kinoshita–Terasaka knot and generalised Conway mutation, J. Knot Theory Ramifications 9 (2000) 557–575