Papers
Topics
Authors
Recent
Search
2000 character limit reached

Small Heegaard genus and SU(2)

Published 18 Sep 2023 in math.GT | (2309.09780v3)

Abstract: Let $Y$ be a closed, orientable 3-manifold with Heegaard genus 2. We prove that if $H_1(Y;\mathbb{Z})$ has order $1$, $3$, or $5$, then there is a representation $\pi_1(Y) \to \mathrm{SU}(2)$ with non-abelian image. Similarly, if $H_1(Y;\mathbb{Z})$ has order $2$ then we find a non-abelian representation $\pi_1(Y) \to \mathrm{SO}(3)$. We also prove that a knot $K$ in $S3$ is a trefoil if and only if there is a unique conjugacy class of irreducible representations $\pi_1(S3\setminus K) \to \mathrm{SU}(2)$ sending a fixed meridian to $\left(\begin{smallmatrix}i&0\0&-i\end{smallmatrix}\right)$.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (46)
  1. Equivariant Floer theory and gluing Donaldson polynomials. Topology, 35(1):167–200, 1996.
  2. Branched covers of quasi-positive links and L-spaces. J. Topol., 12(2):536–576, 2019.
  3. Heegaard splittings of branched coverings of S3superscript𝑆3S^{3}italic_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT. Trans. Amer. Math. Soc., 213:315–352, 1975.
  4. Small Dehn surgery and S⁢U⁢(2)𝑆𝑈2SU(2)italic_S italic_U ( 2 ). arXiv:2110.02874, 2021.
  5. S. Boyer and A. Nicas. Varieties of group representations and Casson’s invariant for rational homology 3333-spheres. Trans. Amer. Math. Soc., 322(2):507–522, 1990.
  6. J. Bowden. Approximating C0superscript𝐶0C^{0}italic_C start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT-foliations by contact structures. Geom. Funct. Anal., 26(5):1255–1296, 2016.
  7. J. A. Baldwin and S. Sivek. Stein fillings and SU(2) representations. Geom. Topol., 22(7):4307–4380, 2018.
  8. J. A. Baldwin and S. Sivek. Instanton L𝐿Litalic_L-spaces and splicing. Ann. H. Lebesgue, 5:1213–1233, 2022.
  9. J. A. Baldwin and S. Sivek. Khovanov homology detects the trefoils. Duke Math. J., 171(4):885–956, 2022.
  10. J. A. Baldwin and S. Sivek. Instantons and L-space surgeries. J. Eur. Math. Soc. (JEMS), 25(10):4033–4122, 2023.
  11. C. R. Cornwell. Character varieties of knot complements and branched double-covers via the cord ring. arXiv:1509.04962, 2015.
  12. S. K. Donaldson. Polynomial invariants for smooth four-manifolds. Topology, 29(3):257–315, 1990.
  13. S. K. Donaldson. Floer homology groups in Yang-Mills theory, volume 147 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2002. With the assistance of M. Furuta and D. Kotschick.
  14. Y. Eliashberg. A few remarks about symplectic filling. Geom. Topol., 8:277–293, 2004.
  15. Confoliations, volume 13 of University Lecture Series. American Mathematical Society, Providence, RI, 1998.
  16. Fixed point-free pseudo-Anosovs and the cinquefoil. arXiv:2203.01402, 2022.
  17. C. M. Herald. Flat connections, the Alexander invariant, and Casson’s invariant. Comm. Anal. Geom., 5(1):93–120, 1997.
  18. F. Hosokawa and S. Kinoshita. On the homology group of branched cyclic covering spaces of links. Osaka Math. J., 12:331–355, 1960.
  19. M. Heusener and E. Klassen. Deformations of dihedral representations. Proc. Amer. Math. Soc., 125(10):3039–3047, 1997.
  20. On transverse knots and branched covers. Int. Math. Res. Not. IMRN, (3):512–546, 2009.
  21. C. Hodgson and J. H. Rubinstein. Involutions and isotopies of lens spaces. In Knot theory and manifolds (Vancouver, B.C., 1983), volume 1144 of Lecture Notes in Math., pages 60–96. Springer, Berlin, 1985.
  22. E. P. Klassen. Representations of knot groups in SU⁢(2)SU2{\rm SU}(2)roman_SU ( 2 ). Trans. Amer. Math. Soc., 326(2):795–828, 1991.
  23. Dehn surgery, the fundamental group and SU(2)2(2)( 2 ). Math. Res. Lett., 11(5-6):741–754, 2004.
  24. Witten’s conjecture and property P. Geom. Topol., 8:295–310, 2004.
  25. P. Kronheimer and T. Mrowka. Instanton Floer homology and the Alexander polynomial. Algebr. Geom. Topol., 10(3):1715–1738, 2010.
  26. P. Kronheimer and T. Mrowka. Knots, sutures, and excision. J. Differential Geom., 84(2):301–364, 2010.
  27. Khovanov homology is an unknot-detector. Publ. Math. Inst. Hautes Études Sci., (113):97–208, 2011.
  28. Monopoles and lens space surgeries. Ann. of Math. (2), 165(2):457–546, 2007.
  29. W. H. Kazez and R. Roberts. C0superscript𝐶0C^{0}italic_C start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT approximations of foliations. Geom. Topol., 21(6):3601–3657, 2017.
  30. Signature of links. Trans. Amer. Math. Soc., 216:351–365, 1976.
  31. Y. Lim. Instanton homology and the Alexander polynomial. Proc. Amer. Math. Soc., 138(10):3759–3768, 2010.
  32. X.-S. Lin. A knot invariant via representation spaces. J. Differential Geom., 35(2):337–357, 1992.
  33. Z. Li and Y. Liang. Some computations on instanton knot homology. J. Knot Theory Ramifications, 32(2):Paper No. 2350007, 14, 2023.
  34. Toroidal integer homology three-spheres have irreducible S⁢U⁢(2)𝑆𝑈2SU(2)italic_S italic_U ( 2 )-representations. J. Topol., 16(1):344–367, 2023.
  35. K. Motegi. Haken manifolds and representations of their fundamental groups in SL⁢(2,𝐂)SL2𝐂{\rm SL}(2,{\bf C})roman_SL ( 2 , bold_C ). Topology Appl., 29(3):207–212, 1988.
  36. K. Murasugi. On a certain numerical invariant of link types. Trans. Amer. Math. Soc., 117:387–422, 1965.
  37. F. Nagasato and Y. Yamaguchi. On the geometry of the slice of trace-free S⁢L2⁢(ℂ)𝑆subscript𝐿2ℂSL_{2}(\mathbb{C})italic_S italic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( blackboard_C )-characters of a knot group. Math. Ann., 354(3):967–1002, 2012.
  38. P. Ozsváth and Z. Szabó. Holomorphic disks and genus bounds. Geom. Topol., 8:311–334, 2004.
  39. L. Rudolph. Some 3-dimensional transverse ℂℂ\mathbb{C}blackboard_C-links (constructions of higher-dimensional ℂℂ\mathbb{C}blackboard_C-links, I). In Interactions between low-dimensional topology and mapping class groups, volume 19 of Geom. Topol. Monogr., pages 367–413. Geom. Topol. Publ., Coventry, 2015.
  40. H. Schubert. Über eine numerische Knoteninvariante. Math. Z., 61:245–288, 1954.
  41. S. Sivek. Donaldson invariants of symplectic manifolds. Int. Math. Res. Not. IMRN, (6):1688–1716, 2015.
  42. S. Sivek and R. Zentner. A menagerie of S⁢U⁢(2)𝑆𝑈2SU(2)italic_S italic_U ( 2 )-cyclic 3-manifolds. Int. Math. Res. Not. IMRN, (11):8038–8085, 2022.
  43. W. P. Thurston. Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Amer. Math. Soc. (N.S.), 6(3):357–381, 1982.
  44. G. Torres. On the Alexander polynomial. Ann. of Math. (2), 57:57–89, 1953.
  45. Y. Xie and B. Zhang. On meridian-traceless SU⁢(2)SU2{\rm SU}(2)roman_SU ( 2 )-representations of link groups. Adv. Math., 418:Paper No. 108947, 48, 2023.
  46. R. Zentner. A class of knots with simple S⁢U⁢(2)𝑆𝑈2SU(2)italic_S italic_U ( 2 )-representations. Selecta Math. (N.S.), 23(3):2219–2242, 2017.

Summary

No one has generated a summary of this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Sign Up to Summarize

Paper to Video (Beta)

Whiteboard

No one has generated a whiteboard explanation for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Sign Up to Generate

Open Problems

We haven't generated a list of open problems mentioned in this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Generate Now

Continue Learning

We haven't generated follow-up questions for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Generate Now

Authors (2)

  1. John A. Baldwin 
  2. Steven Sivek 

Collections

Sign up for free to add this paper to one or more collections.

User Circle Single Streamline Icon: https://streamlinehq.com Sign Up

Tweets

Sign up for free to view the 1 tweet with 0 likes about this paper.

User Circle Single Streamline Icon: https://streamlinehq.com Sign Up for Free