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Flat fully augmented links are determined by their complements (2302.02002v3)
Published 3 Feb 2023 in math.GT
Abstract: In this paper, we show that two flat fully augmented links with homeomorphic complements must be equivalent as links in $\mathbb{S}{3}$. This requires a careful analysis of how totally geodesic surfaces and cusps intersect in these link complements and behave under homeomorphism. One consequence of this analysis is a complete classification of flat fully augmented link complements that admit multiple reflection surfaces. In addition, our work classifies those symmetries of flat fully augmented link complements which are not induced by symmetries of the corresponding link.
- C.C. Adams Augmented alternating link complements are hyperbolic, Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984), London Math. Soc. Lecture Note Ser. 112, Cambridge University Press, Cambridge (1986), 115-130.
- C. C. Adams Thrice-punctured spheres in hyperbolic 3-manifolds, Trans. A. M. S. 287 (1985), no. 2, 645-656. MR 768730 (86k:57008).
- R. Benedetti, C. Petronio Lectures on hyperbolic geometry, Universitext, Springer-Verlag, Berlin (1992).
- J. Berge Embedding the exteriors of one-tunnel knots and links in the 3-sphere unpublished manuscript
- Essential surfaces in highly twisted link complements, Algebr. Geom. Topol. 15 (2015), no. 3, 1501–1523.
- Some virtually special hyperbolic 3-manifold groups, Comment. Math. Helv. 87 (2012), no. 3, 727–787.
- R. Flint, Intercusp Geodesics and Cusp Shapes of Fully Augmented Links, ProQuest LLC, Ann Arbor, MI, 2017, Thesis (Ph.D.) - City University of New York.
- Links with no exceptional surgeries, Comment. Math. Helv. 82 (2007), no. 3, 629–664.
- F. Gainullin, Heegaard Floer homology and knots determined by their complements, Algebr. Geom. Topol. 18 (2018), no. 1 69–109.
- C. McA. Gordon, Links and their complements, Amer. Math. Soc., 314 (2002), 71–82.
- C. McA. Gordon and J. Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989), no. 2, 371-415.
- Symmetry groups of hyperbolic knots and links, Journal of Knot Theory and its Ramifications, 1 (1992), no. 2 185–201.
- Symmetries and Hidden Symmetries of (ϵ,dL)italic-ϵsubscript𝑑𝐿(\epsilon,d_{L})( italic_ϵ , italic_d start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT )-Twisted Knot Complements Algebr. Geom. Topol. 22 (2022), no. 2 601–656.
- Knots in homology spaces determined by their complements, Bull. Korean Math. Soc. 59 (2022), no. 4, 869–877.
- S. Knavel, R.Trapp Embedded totally geodesic surfaces in fully augmented links, Accepted for publication in Comm. in Anal. and Geom.
- M. Lackenby, The volume of hyperbolic alternating link complements, Proc. London Math. Soc. (3) 88 (2004), no. 1, 204–224, With an appendix by Ian Agol and Dylan Thurston.
- B. Mangum, T. Stanford, Brunnian links are determined by their complements, Algebraic &\&& Geometric Topology, 1 (2001), 143–152.
- D. Matignon, On the knot complement problem for non-hyperbolic knots, Topology Appl., 157 (2010), no. 12, 1900-1925.
- Arithmeticity and hidden symmetries of fully augmented pretzel link complements, New York Journal of Math., 26 (2020), 149–183.
- J. Purcell, An introduction to fully augmented links, Interactions between hyperbolic geometry, quantum topology and number theory, 205–220, Contemp. Math., 541, Amer. Math. Soc., Providence, RI, 2011.
- J. Purcell, Cusp shapes under cone deformation, Journal of Differential Geometry, 80 (2008), 453–500.
- J. Purcell, Volumes of highly twisted knots and links, Algebraic & Geometric Topology 7 (2007), 93–108.
- Y.W. Rong, Some knots not determined by their complements, Quantum Topology, 339–353, Ser. Knots Everything, 3, World Sc. Publ., River Edge, NJ. (1993).
- K. Yoshida, Unions of 3-punctured spheres in hyperbolic 3-manifolds, Comm. in Anal. and Geo., vol. 29 (2021) 1643–1689.
- J.H.C. Whitehead, On doubled knots, J. London. Math Soc., vol. 12 (1937) 63–71.
- M. Zevenbergen, Crushtaceans and complements of fully augmented and nested links, Honors Thesis at the University of Rochester, (2021) https://www.sas.rochester.edu/mth/undergraduate/honors-papers-plans.html.