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1. Twist positivity, L-space knots, and concordance

   https://doi.org/10.1007/s00029-024-00996-6  

   Krishna, Siddhi; Morton, Hugh (February 2025, Selecta Mathematica)

   Many well studied knots can be realized as positive braid knots where the braid word contains a positive full twist; we say that such knots are twist positive. Some important families of knots are twist positive, including torus knots, 1-bridge braids, algebraic knots, and Lorenz knots. We prove that if a knot is twist positive, the braid index appears as the third exponent in its Alexander polynomial. We provide a few applications of this result. After observing that most known examples of L-space knots are twist positive, we prove: if K is a twist positive L-space knot, the braid index and bridge index of K agree. This allows us to provide evidence for Baker’s reinterpretation of the slice-ribbon conjecture: that every smooth concordance class contains at most one fibered, strongly quasipositive knot. In particular, we provide the first example of an infinite family of positive braid knots which are distinct in concordance, and where, as g tends to infinity, the number of hyperbolic knots of genus g gets arbitrarily large. Finally, we collect some evidence for a few new conjectures, including the following: the braid and bridge indices agree for any L-space knot. 

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2. n-Bridge braids and the braid index

   https://doi.org/10.1142/S0218216523500906  

   Gollero, Dane; Krishna, Siddhi; Loving, Marissa; Neri, Viridiana; Tahir, Izah; White, Len (November 2023, Journal of Knot Theory and Its Ramifications)

   In this work, we find a closed form formula for the braid index of an [Formula: see text]-bridge braid, a class of positive braid knots which simultaneously generalizes torus knots, 1-bridge braids, and twisted torus knots. Our proof is elementary, effective, and self-contained, and partially recovers work of Birman–Kofman. Along the way, we show that the disparate definitions of twisted torus knots in the literature agree. 

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3. 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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4. 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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5. 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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