More Like this

1. Computing a Link Diagram from its Exterior

   Dunfield, Nathan M. ; Obeidin, Malik ; Rudd, Cameron Gates ( January 2022 , Leibniz international proceedings in informatics)

   Goaoc, Xavier ; Kerber, Michael (Ed.)

   A knot is a circle piecewise-linearly embedded into the 3-sphere. The topology of a knot is intimately related to that of its exterior, which is the complement of an open regular neighborhood of the knot. Knots are typically encoded by planar diagrams, whereas their exteriors, which are compact 3-manifolds with torus boundary, are encoded by triangulations. Here, we give the first practical algorithm for finding a diagram of a knot given a triangulation of its exterior. Our method applies to links as well as knots, allows us to recover links with hundreds of crossings. We use it to find the first diagrams known for 23 principal congruence arithmetic link exteriors; the largest has over 2,500 crossings. Other applications include finding pairs of knots with the same 0-surgery, which relates to questions about slice knots and the smooth 4D Poincaré conjecture.
2. Computing a Link Diagram From Its Exterior

   https://doi.org/10.1007/s00454-023-00533-w  

   Dunfield, Nathan M. ; Obeidin, Malik ; Rudd, Cameron Gates ( August 2023 , Discrete & Computational Geometry)

   A knot is a circle piecewise-linearly embedded into the 3-sphere. The topology of a knot is intimately related to that of its exterior, which is the complement of an open regular neighborhood of the knot. Knots are typically encoded by planar diagrams, whereas their exteriors, which are compact 3-manifolds with torus boundary, are encoded by triangulations. Here, we give the first practical algorithm for finding a diagram of a knot given a triangulation of its exterior. Our method applies to links as well as knots, and allows us to recover links with hundreds of crossings. We use it to find the first diagrams known for 23 principal congruence arithmetic link exteriors; the largest has over 2500 crossings. Other applications include finding pairs of knots with the same 0-surgery, which relates to questions about slice knots and the smooth 4D Poincaré conjecture.
3. Instantons and some concordance invariants of knots

   Mrowka, Tomasz S ( October 2019 , ArXivorg)

   Concordance invariants of knots are derived from the instanton homology groups with local coefficients, as introduced in earlier work of the authors. These concordance invariants include a 1-parameter family of homomorphisms fr , from the knot concordance group to R. Prima facie, these concordance invariants have the potential to provide independent bounds on the genus and number of double points for immersed surfaces with boundary a given knot.
4. Heegaard Floer homology and cosmetic surgeries in $S^3$

   https://doi.org/10.4171/JEMS/1218  

   Hanselman, Jonathan ( March 2022 , Journal of the European Mathematical Society)

   If a knot K in S^3 admits a pair of truly cosmetic surgeries, we show that the surgery slopes are either ±2 or ±1/q for some value of q that is explicitly determined by the knot Floer homology of K. Moreover, in the former case the genus of K must be 2, and in the latter case there is a bound relating q to the genus and the Heegaard Floer thickness of K. As a consequence, we show that the cosmetic crossing conjecture holds for alternating knots (or more generally, Heegaard Floer thin knots) with genus not equal to 2. We also show that the conjecture holds for any knot K for which each prime summand of K has at most 16 crossings; our techniques rule out cosmetic surgeries in this setting except for slopes ±1 and ±2 on a small number of knots, and these remaining examples can be checked by comparing hyperbolic invariants. These results make use of the surgery formula for Heegaard Floer homology, which has already proved to be a powerful tool for obstructing cosmetic surgeries; we get stronger obstructions than previously known by considering the full graded theory. We make use of a new graphicalmore »interpretation of knot Floer homology and the surgery formula in terms of immersed curves, which makes the grading information we need easier to access.« less
5. Four-dimensional aspects of tight contact 3-manifolds

   https://doi.org/10.1073/pnas.2025436118  

   Hedden, Matthew ; Raoux, Katherine ( May 2021 , Proceedings of the National Academy of Sciences)

   We conjecture a four-dimensional characterization of tightness: A contact structure on a 3-manifold Y is tight if and only if a slice-Bennequin inequality holds for smoothly embedded surfaces in$Y\times \left[\mathrm{0,1}\right]$. An affirmative answer to our conjecture would imply an analogue of the Milnor conjecture for torus knots: If a fibered link L induces a tight contact structure on Y, then its fiber surface maximizes the Euler characteristic among all surfaces in$Y\times \left[\mathrm{0,1}\right]$with boundary L. We provide evidence for both conjectures by proving them for contact structures with nonvanishing Ozsváth–Szabó contact invariant.