Universal Formulae for SU Casson Invariants of Knots
Authors:
Hans U. Boden and Andrew Nicas
Journal:
Trans. Amer. Math. Soc. 352 (2000), 31493187
MSC (1991):
Primary 57M25; Secondary 05A19, 14D20, 45G10
Published electronically:
March 24, 2000
MathSciNet review:
1695018
Fulltext PDF Free Access
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Abstract: An Casson invariant of a knot is an integer which can be thought of as an algebraictopological count of the number of characters of representations of the knot group which take a longitude into a given conjugacy class. For fibered knots, these invariants can be characterized as Lefschetz numbers which, for generic conjugacy classes, can be computed using a recursive algorithm of Atiyah and Bott, as adapted by Frohman. Using a new idea to solve the AtiyahBott recursion (as simplified by Zagier), we derive universal formulae which explicitly compute the invariants for all . Our technique is based on our discovery that the generating functions associated to the relevant Lefschetz numbers (and polynomials) satisfy certain integral equations.
 1.
M.
F. Atiyah and R.
Bott, The YangMills equations over Riemann surfaces, Philos.
Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505,
523–615. MR
702806 (85k:14006), http://dx.doi.org/10.1098/rsta.1983.0017
 2.
Hans
U. Boden, Invariants of fibred knots from moduli, Geometric
topology (Athens, GA, 1993) AMS/IP Stud. Adv. Math., vol. 2, Amer.
Math. Soc., Providence, RI, 1997, pp. 259–267. MR 1470731
(98j:57005)
 3.
H. U. Boden, Knot invariants and representation varieties, in preparation, 1999.
 4.
Gerhard
Burde and Heiner
Zieschang, Knots, de Gruyter Studies in Mathematics,
vol. 5, Walter de Gruyter & Co., Berlin, 1985. MR 808776
(87b:57004)
 5.
A. Casson, Lecture notes, MSRI Lectures, Berkeley, 1985.
 6.
Usha
V. Desale and S.
Ramanan, Poincaré polynomials of the variety of stable
bundles, Math. Ann. 216 (1975), no. 3,
233–244. MR 0379497
(52 #402)
 7.
G.
P. Egorychev, Integral representation and the computation of
combinatorial sums, Translations of Mathematical Monographs,
vol. 59, American Mathematical Society, Providence, RI, 1984.
Translated from the Russian by H. H. McFadden; Translation edited by Lev J.
Leifman. MR
736151 (85a:05008)
 8.
Charles
Frohman, Unitary representations of knot groups, Topology
32 (1993), no. 1, 121–144. MR 1204411
(94g:57003), http://dx.doi.org/10.1016/00409383(93)90042T
 9.
C.
D. Frohman and D.
D. Long, Casson’s invariant and surgery on knots, Proc.
Edinburgh Math. Soc. (2) 35 (1992), no. 3,
383–395. MR 1187001
(94a:57012), http://dx.doi.org/10.1017/S0013091500005678
 10.
Charles
Frohman and Andrew
Nicas, An intersection homology invariant for knots in a rational
homology 3sphere, Topology 33 (1994), no. 1,
123–158. MR 1259519
(95c:57010), http://dx.doi.org/10.1016/00409383(94)900396
 11.
G.
Harder and M.
S. Narasimhan, On the cohomology groups of moduli spaces of vector
bundles on curves, Math. Ann. 212 (1974/75),
215–248. MR 0364254
(51 #509)
 12.
G. H. Hardy and E. M. Wright, The Theory of Numbers, (1968) fourth edition, Oxford University Press, London.
 13.
M.
S. Narasimhan and C.
S. Seshadri, Stable and unitary vector bundles on a compact Riemann
surface, Ann. of Math. (2) 82 (1965), 540–567.
MR
0184252 (32 #1725)
 14.
John
Riordan, An introduction to combinatorial analysis, Wiley
Publications in Mathematical Statistics, John Wiley & Sons, Inc., New
York; Chapman & Hall, Ltd., London, 1958. MR 0096594
(20 #3077)
 15.
Don
Zagier, Elementary aspects of the Verlinde formula and of the
HarderNarasimhanAtiyahBott formula, Proceedings of the Hirzebruch
65 Conference on Algebraic Geometry (Ramat Gan, 1993) Israel Math. Conf.
Proc., vol. 9, BarIlan Univ., Ramat Gan, 1996,
pp. 445–462. MR 1360519
(96k:14005)
 1.
 M. Atiyah and R. Bott, The YangMills equations on a Riemann surface, Phil. Trans. Roy. Soc. Lond. A. 308 (1982), 524615. MR 85k:14006
 2.
 H. U. Boden, Invariants of fibred knots from moduli, in Geometric Topology, Ed. W. Kazez, AMS/IP Studies in Advanced Mathematics, vol. 2.1, Amer. Math. Soc., 1997, pp. 259267. MR 98j:57005
 3.
 H. U. Boden, Knot invariants and representation varieties, in preparation, 1999.
 4.
 G. Burde and H Zieschang, Knots, de Gruyter Studies in Mathematics 5, Walter de Gruyter, Berlin, 1985. MR 87b:57004
 5.
 A. Casson, Lecture notes, MSRI Lectures, Berkeley, 1985.
 6.
 U. V. Desale and S. Ramanan, Poincaré polynomials of the variety of stable bundles, Math. Ann. 216 (1975), 233244. MR 52:402
 7.
 G. P. Egorychev, Integral Representation and the Computation of Combinatorial Sums, AMS Translations of Mathematical Monographs, v. 59, Providence (1984). MR 85a:05008
 8.
 C. Frohman, Unitary representations of knot groups, Topology 32 (1993), 121144. MR 94g:57003
 9.
 C. Frohman and D. Long, Casson's invariant and surgery on knots, Proc. Edinburgh Math. Soc. 35 (1992), 383395. MR 94a:57012
 10.
 C. Frohman and A. Nicas, An intersection homology invariant for knots in a rational homology 3sphere, Topology 33 (1994), 123158. MR 95c:57010
 11.
 G. Harder and M. S. Narasimhan, On the cohomology groups of moduli spaces of vector bundles over curves, Math. Ann. 212 (1975), 215248. MR 51:509
 12.
 G. H. Hardy and E. M. Wright, The Theory of Numbers, (1968) fourth edition, Oxford University Press, London.
 13.
 M. Narasimhan and C. Seshadri, Stable and unitary vector bundles on a compact Riemann surface, Ann. Math. 82 (1965), 540567. MR 32:1725
 14.
 J. Riordan, An Introduction to Combinatorial Analysis, John Wiley, New York, 1958. MR 20:3077
 15.
 D. Zagier, Elementary aspects of the Verlinde formula and of the HarderNarasimhanAtiyahBott formula, Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993), 445462, Israel Math. Conf. Proc. 9 BarIlan Univ., Ramat Gan, 1996, pp. 445462. MR 96k:14005
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Additional Information
Hans U. Boden
Affiliation:
Department of Mathematics, Ohio State University, Mansfield, Ohio 44906
Email:
boden@math.ohiostate.edu
Andrew Nicas
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1
Email:
nicas@mcmaster.ca
DOI:
http://dx.doi.org/10.1090/S0002994700025575
PII:
S 00029947(00)025575
Keywords:
Casson invariants,
$\operatorname{SU}(n)$,
fibered knots,
Alexander polynomial,
Conway polynomial,
integral equations
Received by editor(s):
February 20, 1998
Published electronically:
March 24, 2000
Additional Notes:
The secondnamed author was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada
Article copyright:
© Copyright 2000
American Mathematical Society
