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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. 

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2. A Symmetry Motivated Link Table

   https://doi.org/10.3390/sym10110604  

   Witte, Shawn ; Flanner, Michelle ; Vazquez, Mariel ( November 2018 , Symmetry)

   Proper identification of oriented knots and 2-component links requires a precise link nomenclature. Motivated by questions arising in DNA topology, this study aims to produce a nomenclature unambiguous with respect to link symmetries. For knots, this involves distinguishing a knot type from its mirror image. In the case of 2-component links, there are up to sixteen possible symmetry types for each link type. The study revisits the methods previously used to disambiguate chiral knots and extends them to oriented 2-component links with up to nine crossings. Monte Carlo simulations are used to report on writhe, a geometric indicator of chirality. There are ninety-two prime 2-component links with up to nine crossings. Guided by geometrical data, linking number, and the symmetry groups of 2-component links, canonical link diagrams for all but five link types (9 5 2, 9 34 2, 9 35 2, 9 39 2, and 9 41 2) are proposed. We include complete tables for prime knots with up to ten crossings and prime links with up to nine crossings. We also prove a result on the behavior of the writhe under local lattice moves. 

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3. Visualizing Mathematical Knot Equivalence

   https://doi.org/https://doi.org/10.2352/ISSN.2470-1173.2019.1.VDA-683  

   Juan Lin, Hui Zhang ( January 2019 , Electronic Imaging, Visualization and Data Analysis 2019)

   We present a computer interface to visualize and interact with mathematical knots, i.e., the embeddings of closed circles in 3-dimensional Euclidean space. Mathematical knots are slightly different than everyday knots in that they are infinitely stretchy and flexible when being deformed into their topological equivalence. In this work, we design a visualization interface to depict mathematical knots as closed node-link diagrams with energies charged at each node, so that highly-tangled knots can evolve by themselves from high-energy states to minimal (or lower) energy states. With a family of interactive methods and supplementary user interface elements, out tool allows one to sketch, edit, and experiment with mathematical knots, and observe their topological evolution towards optimal embeddings. In addition, out interface can extract from the entire knot evolution those key moments where successive terms in the sequence differ by critical change; this provides a clear and intuitive way to understand and trace mathematical evolution with a minimal number of visual frames. Finally out interface is adapted and extended to support the depiction of mathematical links and braids, whose mathematical concepts and interactions are just similar to our intuition about knots. All these combine to show a mathematically rich interface to help us explore and understand a family of fundamental geometric and topological problems. 

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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 graphical interpretation of knot Floer homology and the surgery formula in terms of immersed curves, which makes the grading information we need easier to access. 

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5. Lombardi drawings of knots and links

   https://doi.org/10.1007/978-3-319-73915-1_10  

   Kindermann, P ; Kobourov, S ; Loeffler, M ; Noellenburg, M ; Schulz, A. ; Vogtenhuber, B ( January 2019 , Journal of computational geometry)

   Knot and link diagrams are projections of one or more 3-dimensional simple closed curves into lR2, such that no more than two points project to the same point in lR2. These diagrams are drawings of 4-regular plane multigraphs. Knots are typically smooth curves in lR3, so their projections should be smooth curves in lR2 with good continuity and large crossing angles: exactly the properties of Lombardi graph drawings (defned by circular-arc edges and perfect angular resolution). We show that several knots do not allow crossing-minimal plane Lombardi drawings. On the other hand, we identify a large class of 4-regular plane multigraphs that do have plane Lombardi drawings. We then study two relaxations of Lombardi drawings and show that every knot admits a crossing-minimal plane 2-Lombardi drawing (where edges are composed of two circular arcs). Further, every knot is near-Lombardi, that is, it can be drawn as a plane Lombardi drawing when relaxing the angular resolution requirement by an arbitrary small angular offset ε, while maintaining a 180◦ angle between opposite edges. 

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