Non-triviality of the A-polynomial for knots in S^3

By Nathan M. Dunfield, Stavros Garoufalidis. 2004

56 citations recorded by the source corpus.

The A-polynomial of a knot in S^3 defines a complex plane curve associated to the set of representations of the fundamental group of the knot exterior into SL(2,C). Here, we show that a non-trivial knot in S^3 has a non-trivial A-polynomial. We deduce this from the gauge-theoretic work of Kronheimer and Mrowka on SU_2-representations of Dehn surgeries on knots in S^3. As a corollary, we show that if a conjecture connecting the colored Jones polynomials to the A-polynomial holds, then the colored Jones polynomials distinguish the unknot.

Subfields: Genetics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bf04b7d1f511157923c6b5957f3ee56b2c23d22f

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In Genetics

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4. Michael C. Sullivan
5. Tudor Dimofte
6. O. Plamenevskaya
7. Peter S. Ozsv'ath
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In Geometry and Topology

1. T. Ekholm
2. John B. Etnyre
3. Lenhard L. Ng
4. Michael C. Sullivan
5. Georgios Dimitroglou Rizell
6. Tudor Dimofte
7. Tam'as K'alm'an
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In Mathematical Physics

1. John B. Etnyre
2. T. Ekholm
3. Michael C. Sullivan
4. Lenhard L. Ng
5. Tudor Dimofte
6. Georgios Dimitroglou Rizell
7. Michael Hutchings
8. A. Stipsicz

Citations in the corpus

• Quantum Riemann Surfaces in Chern-Simons Theory (2011, background): “It is known that any nontrivial knot in S has a nontrivial A-polynomial; in other words, there are always components besides (`− 1) [49].”
• Knot contact homology (2011, background): “By a result (Theorem 4.3) whose proof is deferred to Section 5, there is a one-to-one correspondence between rigid multiscale flow trees and rigid holomorphic disks with boundary on ΛK and one positive puncture.”