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1. Crossing numbers of cable knots

   https://doi.org/10.1112/blms.13140  

   Kalfagianni, Efstratia; Mcconkey, Rob (June 2024, Bulletin of the London Mathematical Society)

   We use the degree of the colored Jones knot polynomials to show that the crossing number of a (p,q)‐cable of an adequate knot with crossing number c is larger than q^2 c. As an application, we determine the crossing number of 2‐cables of adequate knots. We also determine the crossing number of the connected sum of any adequate knot with a 2‐cable of an adequate knot. 

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2. Positive braids minimize ascending number

   https://doi.org/10.1142/S0218216525500725  

   Davis, Lowell; Meier, Jeffrey (January 2026, Journal of Knot Theory and Its Ramifications)

   A well-known algorithm for unknotting knots involves traversing a knot diagram and changing each crossing that is first encountered from below. The minimal number of crossings changed in this way across all diagrams for a knot is called the ascending number of the knot. The ascending number is bounded below by the unknotting number. We show that for knots obtained as the closure of a positive braid, the ascending number equals the unknotting number. We also present data indicating that a similar result may hold for positive knots. We use this data to examine which low-crossing knots have the property that their ascending number is realized in a minimal crossing diagram, showing that there are at most 5 hyperbolic, alternating knots with at most 12 crossings with this property. 

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3. The distribution of knots in the Petaluma model

   https://doi.org/10.2140/agt.2018.18.3647  

   Even-Zohar, C; Hass, J; Linial, N; Nowik, T (October 2018, Algebraic geometric topology)

   The representation of knots by petal diagrams (Adams et al 2012) naturally defines a sequence of distributions on the set of knots. We establish some basic properties of this randomized knot model. We prove that in the random n–petal model the probability of obtaining every specific knot type decays to zero as n, the number of petals, grows. In addition we improve the bounds relating the crossing number and the petal number of a knot. This implies that the n–petal model represents at least exponentially many distinct knots. Past approaches to showing, in some random models, that individual knot types occur with vanishing probability rely on the prevalence of localized connect summands as the complexity of the knot increases. However, this phenomenon is not clear in other models, including petal diagrams, random grid diagrams and uniform random polygons. Thus we provide a new approach to investigate this question. 

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4. Ribbon numbers of 12-crossing knots

   https://doi.org/10.1142/S021821652550066X  

   An, Xianhao; Aronin, Matthew; Cates, David; Goh, Ansel; Kirn, Benjamin; Krienke, Josh; Liang, Minyi; Lowery, Samuel; Malkoc, Ege; Meier, Jeffrey; et al (November 2025, Journal of Knot Theory and Its Ramifications)

   The ribbon number of a knot is the minimum number of ribbon singularities among all ribbon disks bounded by that knot. In this paper, we build on the systematic treatment of this knot invariant initiated in recent work of Friedl, Misev, and Zupan. We show that the set of Alexander polynomials of knots with ribbon number at most four contains 56 polynomials, and we use this set to compute the ribbon numbers for many 12-crossing knots. We also study higher-genus ribbon numbers of knots, presenting some examples that exhibit interesting behavior and establishing that the success of the Alexander polynomial at controlling genus-0 ribbon numbers does not extend to higher genera. 

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5. Knot Floer homology and the unknotting number

   Alishahi, A.; Eftekhary, E. (January 2021, Geometry topology)

   Given a knot K in S3, let u−(K) (respectively, u+(K)) denote the minimum number of negative (respectively, positive) crossing changes among all unknotting sequences for K. We use knot Floer homology to construct the invariants l−(K),l+(K) and l(K), which give lower bounds on u−(K),u+(K) and the unknotting number u(K), respectively. The invariant l(K) only vanishes for the unknot, and satisfies l(K) ≥ ν+(K), while the difference l(K) − ν+(K) can be arbitrarily large. We also present several applications towards bounding the unknotting number, the alteration number and the Gordian distance. 

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