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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. Knots with exactly 10 sticks

   https://doi.org/10.1142/S021821652050011X  

   Blair, Ryan; Eddy, Thomas D.; Morrison, Nathaniel; Shonkwiler, Clayton (March 2020, Journal of Knot Theory and Its Ramifications)

   We prove that the knots [Formula: see text] and [Formula: see text] both have stick number 10. These are the first non-torus prime knots with more than 9 crossings for which the exact stick number is known. 

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3. Fillable contact structures from positive surgery

   https://doi.org/10.1090/btran/200  

   Mark, Thomas; Tosun, Bülent (August 2024, Transactions of the American Mathematical Society, Series B)

   We consider the question of when the operation of contact surgery with positive surgery coefficient, along a knot $K$in a contact 3-manifold $Y$, gives rise to a weakly fillable contact structure. We show that this happens if and only if $Y$itself is weakly fillable, and $K$is isotopic to the boundary of a properly embedded symplectic disk inside a filling of $Y$. Moreover, if $Y^{\prime }$is a contact manifold arising from positive contact surgery along $K$, then any filling of $Y^{\prime }$is symplectomorphic to the complement of a suitable neighborhood of such a disk in a filling of $Y$. Using this result we deduce several necessary conditions for a knot in the standard 3-sphere to admit a fillable positive surgery, such as quasipositivity and equality between the slice genus and the 4-dimensional clasp number, and we give a characterization of such knots in terms of a quasipositive braid expression. We show that knots arising as the closure of a positive braid always admit a fillable positive surgery, as do knots that have lens space surgeries, and suitable satellites of such knots. In fact the majority of quasipositive knots with up to 10 crossings admit a fillable positive surgery. On the other hand, in general, (strong) quasipositivity, positivity, or Lagrangian fillability need not imply a knot admits a fillable positive contact surgery. 

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4. Meridional rank and bridge number of knotted 2-spheres

   https://doi.org/10.4153/S0008414X23000883  

   Joseph, Jason; Pongtanapaisan, Puttipong (February 2025, Canadian Journal of Mathematics)

   Abstract The meridional rank conjecture asks whether the bridge number of a knot in$$S^3$$is equal to the minimal number of meridians needed to generate the fundamental group of its complement. In this paper, we investigate the analogous conjecture for knotted spheres in$$S^4$$. Towards this end, we give a construction to produce classical knots with quotients sending meridians to elements of any finite order in Coxeter groups and alternating groups, which detect their meridional ranks. We establish the equality of bridge number and meridional rank for these knots and knotted spheres obtained from them by twist-spinning. On the other hand, we show that the meridional rank of knotted spheres is not additive under connected sum, so that either bridge number also collapses, or meridional rank is not equal to bridge number for knotted spheres. 

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5. 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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