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1. Unknotted curves on genus-one Seifert surfaces of Whitehead doubles

   https://doi.org/10.2140/pjm.2024.330.123  

   Dey, Subhankar; King, Veronica; Shaw, Colby T; Tosun, Bülent; Trace, Bruce (January 2024, Pacific Journal of Mathematics)

   We consider homologically essential simple closed curves on Seifert surfaces of genus one knots in S3, and in particular those that are unknotted or slice in S3. We completely characterize all such curves for most twist knots: they are either positive or negative braid closures; moreover, we determine exactly which of those are unknotted. A surprising consequence of our work is that the figure eight knot admits infinitely many unknotted essential curves up to isotopy on its genus one Seifert surface, and those curves are enumerated by Fibonacci numbers. On the other hand, we prove that many twist knots admit homologically essential curves that cannot be positive or negative braid closures. Indeed, among those curves, we exhibit an example of a slice but not unknotted homologically essential simple closed curve. We further investigate our study of unknotted essential curves for arbitrary Whitehead doubles of non-trivial knots, and obtain that there is a precisely one unknotted essential simple closed curve in the interior of the doubles’ standard genus one Seifert surface. As a consequence of all these we obtain many new examples of 3-manifolds that bound contractible 4-manifolds. 

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2. Splicing knot complements and bordered Floer homology

   https://doi.org/10.1515/crelle-2014-0064  

   Hedden, Matthew; Levine, Adam Simon (January 2016, Journal für die reine und angewandte Mathematik (Crelles Journal))

   null (Ed.)

   Abstract We show that the integer homology sphere obtained by splicing two nontrivial knot complements in integer homology sphere L-spaces has Heegaard Floer homology of rank strictly greater than one. In particular, splicing the complements of nontrivial knots in the 3-sphere never produces an L-space. The proof uses bordered Floer homology. 

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3. Totally Geodesic Surfaces in Hyperbolic 3-Manifolds: Algorithms and Examples

   https://doi.org/10.4230/LIPIcs.SoCG.2024.14  

   Basilio, Brannon; Lee, Chaeryn; Malionek, Joseph (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Mulzer, Wolfgang; Phillips, Jeff M (Ed.)

   Finding a totally geodesic surface, an embedded surface where the geodesics in the surface are also geodesics in the surrounding manifold, has been a problem of interest in the study of 3-manifolds. This has especially been of interest in hyperbolic 3-manifolds and knot complements, complements of piecewise-linearly embedded circles in the 3-sphere. This is due to Menasco-Reid’s conjecture stating that hyperbolic knot complements do not contain such surfaces. Here, we present an algorithm that determines whether a given surface is totally geodesic and an algorithm that checks whether a given 3-manifold contains a totally geodesic surface. We applied our algorithm on over 150,000 3-manifolds and discovered nine 3-manifolds with totally geodesic surfaces. Additionally, we verified Menasco-Reid’s conjecture for knots up to 12 crossings. 

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4. Cusp volumes of alternating knots on surfaces

   https://doi.org/10.2140/agt.2022.22.2467  

   Bavier, B. (November 2022, Algebraic geometric topology)

   We study the geometry of hyperbolic knots that admit alternating projections on embedded surfaces in closed 3–manifolds. We show that, under mild hypothesis, their cusp area admits two-sided bounds in terms of the twist number of the alternating projection and the genus of the projection surface. As a result, we derive diagrammatic estimates of slope lengths and give applications to Dehn surgery. These generalize results of Lackenby and Purcell about alternating knots in the 3–sphere. Using a result of Kalfagianni and Purcell, we point out that alternating knots on surfaces of higher genus can have arbitrarily small cusp density, in contrast to alternating knots on spheres whose cusp densities are bounded away from zero due to Lackenby and Purcell. 

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5. Representation Homology of Topological Spaces

   https://doi.org/10.1093/imrn/rnaa023  

   Berest, Yuri; Ramadoss, Ajay C; Yeung, Wai-Kit (February 2020, International Mathematics Research Notices)

   null (Ed.)

   Abstract In this paper, we introduce and study representation homology of topological spaces, which is a natural homological extension of representation varieties of fundamental groups. We give an elementary construction of representation homology parallel to the Loday–Pirashvili construction of higher Hochschild homology; in fact, we establish a direct geometric relation between the two theories by proving that the representation homology of the suspension of a (pointed connected) space is isomorphic to its higher Hochschild homology. We also construct some natural maps and spectral sequences relating representation homology to other homology theories associated with spaces (such as Pontryagin algebras, $${{\mathbb{S}}}^1$$-equivariant homology of the free loop space, and stable homology of automorphism groups of f.g. free groups). We compute representation homology explicitly (in terms of known invariants) in a number of interesting cases, including spheres, suspensions, complex projective spaces, Riemann surfaces, and some 3-dimensional manifolds, such as link complements in $${\mathbb{R}}^3$$ and the lens spaces $ L(p,q) $. In the case of link complements, we identify the representation homology in terms of ordinary Hochschild homology, which gives a new algebraic invariant of links in $${\mathbb{R}}^3$$. 

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