|
The number of knot group representations
is not a Vassiliev invariant
Author:
Michael Eisermann
Journal:
Proc. Amer. Math. Soc. 128 (2000), 1555-1561
MSC (1991):
Primary 57M25
Posted:
October 5, 1999
MathSciNet review:
1657727
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: For a finite group  and a knot  in the  -sphere, let  be the number of representations of the knot group into  . In answer to a question of D.Altschuler we show that  is either constant or not of finite type. Moreover,  is constant if and only if  is nilpotent. We prove the following, more general boundedness theorem: If a knot invariant  is bounded by some function of the braid index, the genus, or the unknotting number, then  is either constant or not of finite type.
- 1.
Daniel
Altschuler, Representations of knot groups and Vassiliev
invariants, J. Knot Theory Ramifications 5 (1996),
no. 4, 421–425. MR 1406713
(98c:57001), http://dx.doi.org/10.1142/S0218216596000254
- 2.
Dror
Bar-Natan, On the Vassiliev knot invariants, Topology
34 (1995), no. 2, 423–472. MR 1318886
(97d:57004), http://dx.doi.org/10.1016/0040-9383(95)93237-2
- 3.
Dror
Bar-Natan, Polynomial invariants are polynomial, Math. Res.
Lett. 2 (1995), no. 3, 239–246. MR 1338783
(96c:57006)
- 4.
D.
A. Slavnov, An electromagnetic field without indefinite
metric, Teoret. Mat. Fiz. 29 (1976), no. 3,
291–299 (Russian, with English summary). MR 0471754
(57 #11477)
- 5.
Gerhard
Burde and Heiner
Zieschang, Knots, de Gruyter Studies in Mathematics,
vol. 5, Walter de Gruyter & Co., Berlin, 1985. MR 808776
(87b:57004)
- 6.
John
Dean, Many classical knot invariants are not Vassiliev
invariants, J. Knot Theory Ramifications 3 (1994),
no. 1, 7–10. MR 1265449
(94k:57008), http://dx.doi.org/10.1142/S0218216594000034
- 7.
F.
González-Acu na, Homomorphs of knot groups, Ann. of
Math. (2) 102 (1975), no. 2, 373–377. MR 0379671
(52 #576)
- 8.
B.
Huppert, Endliche Gruppen. I, Die Grundlehren der
Mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin, 1967
(German). MR
0224703 (37 #302)
- 9.
Dennis
Johnson, Homomorphs of knot groups, Proc. Amer. Math. Soc. 78 (1980), no. 1, 135–138. MR 548101
(80j:57004), http://dx.doi.org/10.1090/S0002-9939-1980-0548101-9
- 10.
S.Lambropoulou, C.P.Rourke: Markov's theorem in
 -manifolds. Topology Appl. 78 (1997), Special issue on braid groups and related topics, 95-122 CMP 97:16
- 11.
L.
P. Neuwirth, Knot groups, Annals of Mathematics Studies, No.
56, Princeton University Press, Princeton, N.J., 1965. MR 0176462
(31 #734)
- 12.
Rolland
Trapp, Twist sequences and Vassiliev invariants, J. Knot
Theory Ramifications 3 (1994), no. 3, 391–405.
Random knotting and linking (Vancouver, BC, 1993). MR 1291867
(95h:57012), http://dx.doi.org/10.1142/S0218216594000289
- 13.
Shuji
Yamada, The minimal number of Seifert circles equals the braid
index of a link, Invent. Math. 89 (1987), no. 2,
347–356. MR
894383 (88f:57015), http://dx.doi.org/10.1007/BF01389082
- 14.
M.Zorn: Nilpotency of finite groups. Bull. Amer. Math. Soc. 42 (1936), 485-486
- 1.
- D.Altschuler: Representation of knot groups and Vassiliev invariants. J. Knot Theory Ramifications 5 (1996), 421-425 MR 98c:57001
- 2.
- D.Bar-Natan: On the Vassiliev knot invariants. Topology 34 (1995), 423-472 MR 97d:57004
- 3.
- D.Bar-Natan: Polynomial invariants are polynomial. Math. Res. Lett. 2 (1995), 239-246 MR 96c:57006
- 4.
- J.S.Birman: Braids, links, and mapping class groups. Annals of Mathematics Studies, No.82, Princeton University Press, Princeton 1974 MR 57:11477
- 5.
- G.Burde, H.Zieschang: Knots. de Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin 1985 MR 87b:57004
- 6.
- J.Dean: Many classical knot invariants are not Vassiliev invariants. J. Knot Theory Ramifications 3 (1994), 7-9 MR 94k:57008
- 7.
- F.Gonzalez-Acuña: Homomorphs of knot groups. Ann. of Math. (2) 102 (1975), 373-377 MR 52:576
- 8.
- B.Huppert: Endliche Gruppen I. Grundlehren der Mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin 1967 MR 37:302
- 9.
- D.Johnson: Homomorphs of knot groups. Proc. Amer. Math. Soc. 78 (1980), 135-138 MR 80j:57004
- 10.
- S.Lambropoulou, C.P.Rourke: Markov's theorem in -manifolds. Topology Appl. 78 (1997), Special issue on braid groups and related topics, 95-122 CMP 97:16
- 11.
- L.P.Neuwirth: Knot groups. Annals of Mathematics Studies, No.56, Princeton University Press, Princeton 1965 MR 31:734
- 12.
- R.Trapp: Twist sequences and Vassiliev invariants. J. Knot Theory Ramifications 3 (1994), 391-405 MR 95h:57012
- 13.
- S.Yamada: The minimal number of Seifert circles equals the braid index of a link. Invent. Math. 89 (1987), 347-356. MR 88f:57015
- 14.
- M.Zorn: Nilpotency of finite groups. Bull. Amer. Math. Soc. 42 (1936), 485-486
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (1991):
57M25
Retrieve articles in all journals
with MSC (1991):
57M25
Additional Information
Michael Eisermann
Affiliation:
Mathematisches Institut der Universität Bonn, Beringstr.1, 53115 Bonn, Germany
Email:
eiserm@math.uni-bonn.de
DOI:
http://dx.doi.org/10.1090/S0002-9939-99-05287-9
PII:
S 0002-9939(99)05287-9
Received by editor(s):
July 9, 1998
Posted:
October 5, 1999
Communicated by:
Ronald A. Fintushel
Article copyright:
© Copyright 2000 American Mathematical Society
|
