Estimating the states of the Kauffman bracket skein module

Doug Bullock

Banach Center Publications (1998)

  • Volume: 42, Issue: 1, page 23-28
  • ISSN: 0137-6934

Abstract

top
The states of the title are a set of knot types which suffice to create a generating set for the Kauffman bracket skein module of a manifold. The minimum number of states is a topological invariant, but quite difficult to compute. In this paper we show that a set of states determines a generating set for the ring of S L 2 ( C ) characters of the fundamental group, which in turn provides estimates of the invariant.

How to cite

top

Bullock, Doug. "Estimating the states of the Kauffman bracket skein module." Banach Center Publications 42.1 (1998): 23-28. <http://eudml.org/doc/208809>.

@article{Bullock1998,
abstract = {The states of the title are a set of knot types which suffice to create a generating set for the Kauffman bracket skein module of a manifold. The minimum number of states is a topological invariant, but quite difficult to compute. In this paper we show that a set of states determines a generating set for the ring of $SL_2(C)$ characters of the fundamental group, which in turn provides estimates of the invariant.},
author = {Bullock, Doug},
journal = {Banach Center Publications},
language = {eng},
number = {1},
pages = {23-28},
title = {Estimating the states of the Kauffman bracket skein module},
url = {http://eudml.org/doc/208809},
volume = {42},
year = {1998},
}

TY - JOUR
AU - Bullock, Doug
TI - Estimating the states of the Kauffman bracket skein module
JO - Banach Center Publications
PY - 1998
VL - 42
IS - 1
SP - 23
EP - 28
AB - The states of the title are a set of knot types which suffice to create a generating set for the Kauffman bracket skein module of a manifold. The minimum number of states is a topological invariant, but quite difficult to compute. In this paper we show that a set of states determines a generating set for the ring of $SL_2(C)$ characters of the fundamental group, which in turn provides estimates of the invariant.
LA - eng
UR - http://eudml.org/doc/208809
ER -

References

top
  1. [1] G. Brumfiel and H. M. Hilden, SL(2) representations of finitely presented groups, Contemporary Mathematics 187 (1995). Zbl0838.20006
  2. [2] D. Bullock, The (2,∞)-skein module of the complement of a (2,2p+1) torus knot, J. Knot Theory Ramifications 4 no. 4 (1995) 619-632. Zbl0852.57003
  3. [3] D. Bullock, On the Kauffman bracket skein module of surgery on a trefoil, Pacific J. Math., to appear. Zbl0878.57005
  4. [4] D. Bullock, A finite set of generators for the Kauffman bracket skein algebra, preprint. Zbl0932.57016
  5. [5] D. Bullock, Estimating a skein module with S L 2 ( C ) characters, Proc. Amer. Math. Soc., to appear. Zbl0866.57005
  6. [6] M. Culler and P. Shalen, Varieties of group representations and splittings of 3-manifolds, Ann. Math. 117 (1983) 109-146. Zbl0529.57005
  7. [7] R. Fricke and F. Klein, Vorlesungen über die Theorie der automorphen Functionen, Vol. 1, B. G. Teubner, Leipzig 1897. 
  8. [8] W. Goldman, The Symplectic Nature of Fundamental Groups of Surfaces, Adv. Math. 54 no. 2 (1984) 200-225. Zbl0574.32032
  9. [9] R. Horowitz, Characters of free groups represented in the two dimensional linear group, Comm. Pure Appl. Math. 25 (1972) 635-649. Zbl1184.20009
  10. [10] J. Hoste and J. H. Przytycki, The (2,∞)-skein module of lens spaces; a generalization of the Jones polynomial, J. Knot Theory Ramifications 2 no. 3 (1993) 321-333. Zbl0796.57005
  11. [11] J. Hoste and J. H. Przytycki, The Kauffman bracket skein module of S 1 × S 2 , Math Z. 220 (1995) 65-73. Zbl0826.57007
  12. [12] W. Magnus, Rings of Fricke characters and automorphism groups of free groups, Math. Z. 170 (1980), 91-103. Zbl0433.20033
  13. [13] H. Vogt, Sur les invariants fondamentaux des équations différentielles linéaires du second ordre, Ann. Sci. École Norm. Supér. III. Sér. 6 (1889), 3-72. Zbl21.0314.01

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.