TY - JOUR
T1 - Understanding the Kauffman bracket skein module
AU - Bullock, Doug
AU - Frohman, Charles
AU - Kania-Bartoszyńska, Joanna
PY - 1999/5
Y1 - 1999/5
N2 - The Kauffman bracket skein module K(M) of a 3-manifold M is defined over formal power series in the variable h by letting A = eh/4. For a compact oriented surface F, it is shown that K(F×I) is a quantization of the SL2(ℂ)-characters of the fundamental group of F corresponding to a geometrically defined Poisson bracket. Finite type invariants for unoriented knots and links are defined and obtained from topologically free Kauffman bracket modules. A structure theorem for K(M) is given in terms of the affine SL2(ℂ)-characters of π1(M). It follows for compact M that K(M) can be generated as a module by cables on a finite set of knots. Moreover, if M contains no incompressible surfaces, the module is topologically finitely generated.
AB - The Kauffman bracket skein module K(M) of a 3-manifold M is defined over formal power series in the variable h by letting A = eh/4. For a compact oriented surface F, it is shown that K(F×I) is a quantization of the SL2(ℂ)-characters of the fundamental group of F corresponding to a geometrically defined Poisson bracket. Finite type invariants for unoriented knots and links are defined and obtained from topologically free Kauffman bracket modules. A structure theorem for K(M) is given in terms of the affine SL2(ℂ)-characters of π1(M). It follows for compact M that K(M) can be generated as a module by cables on a finite set of knots. Moreover, if M contains no incompressible surfaces, the module is topologically finitely generated.
KW - 3-manifold
KW - Character theory
KW - Knot
KW - Link
KW - Skein module
UR - http://www.scopus.com/inward/record.url?scp=0033455237&partnerID=8YFLogxK
U2 - 10.1142/S0218216599000183
DO - 10.1142/S0218216599000183
M3 - Article
AN - SCOPUS:0033455237
SN - 0218-2165
VL - 8
SP - 265
EP - 277
JO - Journal of Knot Theory and its Ramifications
JF - Journal of Knot Theory and its Ramifications
IS - 3
ER -