Nontriviality of the 𝑀-degree of the 𝐴-polynomial

Boden, Hans

doi:10.1090/S0002-9939-2014-11936-8

Nontriviality of the $M$-degree of the $A$-polynomial
HTML articles powered by AMS MathViewer

by Hans U. Boden

Proc. Amer. Math. Soc. 142 (2014), 2173-2177

DOI: https://doi.org/10.1090/S0002-9939-2014-11936-8

Published electronically: March 4, 2014

PDF | Request permission

Abstract:

This note gives a proof that the $A$-polynomial of any nontrivial knot in $S^3$ has nontrivial $M$-degree.

References

Hans U. Boden and Cynthia L. Curtis, The $SL(2,\Bbb C)$ Casson invariant for Dehn surgeries on two-bridge knots, Algebr. Geom. Topol. 12 (2012), no. 4, 2095–2126. MR 3020202, DOI 10.2140/agt.2012.12.2095
Steven Boyer and Xingru Zhang, A proof of the finite filling conjecture, J. Differential Geom. 59 (2001), no. 1, 87–176. MR 1909249
Steven Boyer and Xingru Zhang, Every nontrivial knot in $S^3$ has nontrivial $A$-polynomial, Proc. Amer. Math. Soc. 133 (2005), no. 9, 2813–2815. MR 2146231, DOI 10.1090/S0002-9939-05-07814-7
J. C. Cha and C. Livingston, KnotInfo: Table of Knot Invariants, online at http://www.indiana.edu/$\sim$knotinfo, January 31, 2012.
D. Cooper, M. Culler, H. Gillet, D. D. Long, and P. B. Shalen, Plane curves associated to character varieties of $3$-manifolds, Invent. Math. 118 (1994), no. 1, 47–84. MR 1288467, DOI 10.1007/BF01231526
D. Cooper and D. D. Long, Remarks on the $A$-polynomial of a knot, J. Knot Theory Ramifications 5 (1996), no. 5, 609–628. MR 1414090, DOI 10.1142/S0218216596000357
D. Cooper and D. D. Long, The $A$-polynomial has ones in the corners, Bull. London Math. Soc. 29 (1997), no. 2, 231–238. MR 1426004, DOI 10.1112/S0024609396002251
Marc Culler and Peter B. Shalen, Varieties of group representations and splittings of $3$-manifolds, Ann. of Math. (2) 117 (1983), no. 1, 109–146. MR 683804, DOI 10.2307/2006973
Cynthia L. Curtis, An intersection theory count of the $\textrm {SL}_2({\Bbb C})$-representations of the fundamental group of a $3$-manifold, Topology 40 (2001), no. 4, 773–787. MR 1851563, DOI 10.1016/S0040-9383(99)00083-X
Nathan M. Dunfield and Stavros Garoufalidis, Non-triviality of the $A$-polynomial for knots in $S^3$, Algebr. Geom. Topol. 4 (2004), 1145–1153. MR 2113900, DOI 10.2140/agt.2004.4.1145
Stavros Garoufalidis and Thomas W. Mattman, The $A$-polynomial of the $(-2,3,3+2n)$ pretzel knots, New York J. Math. 17 (2011), 269–279. MR 2811064
Jim Hoste and Patrick D. Shanahan, A formula for the A-polynomial of twist knots, J. Knot Theory Ramifications 13 (2004), no. 2, 193–209. MR 2047468, DOI 10.1142/S0218216504003081
M. Ishikawa, T. Mattman, and K. Shimokawa, Tangle sums and factorization of $A$-polynomials, 2011 preprint, arXiv: math.GT 1107.2640.
P. B. Kronheimer and T. S. Mrowka, Dehn surgery, the fundamental group and SU$(2)$, Math. Res. Lett. 11 (2004), no. 5-6, 741–754. MR 2106239, DOI 10.4310/MRL.2004.v11.n6.a3
Thomas W. Mattman, The Culler-Shalen seminorms of the $(-2,3,n)$ pretzel knot, J. Knot Theory Ramifications 11 (2002), no. 8, 1251–1289. MR 1949779, DOI 10.1142/S0218216502002232
K. Petersen, $A$-polynomials of a family of two-bridge knots, 2012 preprint.
Naoko Tamura and Yoshiyuki Yokota, A formula for the $A$-polynomials of $(-2,3,1+2n)$-pretzel knots, Tokyo J. Math. 27 (2004), no. 1, 263–273. MR 2060090, DOI 10.3836/tjm/1244208490
Peter B. Shalen, Representations of 3-manifold groups, Handbook of geometric topology, North-Holland, Amsterdam, 2002, pp. 955–1044. MR 1886685
W. A. Stein et al., Sage Mathematics Software (Version 4.8), The Sage Development Team, 2012, http://www.sagemath.org.