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Spaces of knots in the solid torus, knots in the thickened torus, and links in the 3-sphere

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Abstract

We recursively determine the homotopy type of the space of any irreducible framed link in the 3-sphere, modulo rotations. This leads us to the homotopy type of the space of any knot in the solid torus, thus answering a question posed by Arnold. We similarly study spaces of unframed links in the 3-sphere, modulo rotations, and spaces of knots in the thickened torus. The subgroup of meridional rotations splits as a direct factor of the fundamental group of the space of any framed link except the unknot. Its generators can be viewed as generalizations of the Gramain loop in the space of long knots. Taking the quotient by certain such rotations relates the spaces we study. All of our results generalize previous work of Hatcher and Budney. We provide many examples and explicitly describe generators of fundamental groups.

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Notes

  1. Up to 6 crossings, the number increases from 7 to 526 [22]. However, many of the examples we consider fall outside this range of crossing numbers, especially those involving satellite operations.

  2. The second author credits Rob Kusner for essentially pointing out this connection to him in 2012.

  3. Beware that the space of closed knots (respectively framed closed knots) is a subspace of \({\mathcal {L}}\) (respectively \(\widetilde{{\mathcal {L}}}\)), not \({\mathcal {K}}\) (respectively \(\widetilde{{\mathcal {K}}}\)). Our notation is not overloaded because we make no use of long links.

  4. Note that we are only using the connectedness of this space of diffeomorphisms, which can be deduced from Cerf’s result \(\pi _0\text {Diff}^+(S^3)=\{e\}\), obtained well before the proof of the Smale conjecture.

  5. Indeed, the linking number Lk of \(T_{p',q'}\) with a normal perturbation of \(T_{p',q'}\) on the torus satisfies \(Lk=Tw+Wr\), where Tw is twist and Wr is writhe [12, 52]. By writing \(T_{p',q'}\) as the closure of the appropriate \(p'\)-strand braid, \(Wr=q'(p'-1)\), the number of (positive) crossings. The twist Tw is the linking number with \(C_1\), which is \(q'\).

  6. The reader may find it amusing to realize \(g_F\) using a knotted belt with anchored ends.

  7. This \(D_2<D_4\) is of course the subgroup containing the \(180^\circ \) rotation and reflections across the coordinate axes. The isometry of M corresponding to a reflection of \({\mathbb {R}}^3\) (and an isotopy between mirror images of the knot) can be identified with a reflection across a diagonal in \(D_4\).

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Acknowledgements

The first author thanks R. Inanç Baykur for support and encouragement in participating in this project. The second author thanks Ryan Budney for conservations about approaches to this question, Sam Nariman for a conversation about reducible 3-manifolds, and Rafał Komendarczyk for a brief discussion of connections to other problems. Both authors thank Nikolay Buskin and Richard Buckman for bringing to our attention the problem of Arnold, for useful early conversations, and for inspiration to explore various examples. We thank the referee for useful comments which lead to some strengthening of our results. We acknowledge the use of KLO [58] and especially SnapPy [13] for verifying the symmetries of various links, and the use of Inkscape and Grapher for producing graphics. The second author was supported by the Louisiana Board of Regents Support Fund, contract number LEQSF(2019-22)-RD-A-22.

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  1. Department of Mathematics & Statistics, University of Massachusetts Amherst, Amherst, MA, 01003, USA

    Andrew Havens

  2. Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA, 70504, USA

    Robin Koytcheff

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  1. Andrew Havens
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Havens, A., Koytcheff, R. Spaces of knots in the solid torus, knots in the thickened torus, and links in the 3-sphere. Geom Dedicata 214, 671–737 (2021). https://doi.org/10.1007/s10711-021-00633-y

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