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1. Twist number and the alternating volume of knots

   https://doi.org/10.1142/S0218216519500160  

   Blair, Ryan; Allen, Heidi; Rodriguez, Leslie (January 2019, Journal of Knot Theory and Its Ramifications)

   It was previously shown by the first author that every knot in [Formula: see text] is ambient isotopic to one component of a two-component, alternating, hyperbolic link. In this paper, we define the alternating volume of a knot [Formula: see text] to be the minimum volume of any link [Formula: see text] in a natural class of alternating, hyperbolic links such that [Formula: see text] is ambient isotopic to a component of [Formula: see text]. Our main result shows that the alternating volume of a knot is coarsely equivalent to the twist number of a knot. 

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2. The non-orientable 4-genus of 11 crossing non-alternating knots

   https://doi.org/10.1142/S0218216524500500  

   Fairchild, Megan (January 2025, Journal of Knot Theory and Its Ramifications)

   The non-orientable 4-genus of a knot [Formula: see text] in [Formula: see text] is defined to be the minimum first Betti number of a non-orientable surface [Formula: see text] smoothly embedded in [Formula: see text] so that [Formula: see text] bounds [Formula: see text]. We will survey the tools used to compute the non-orientable 4-genus, and use various techniques to calculate this invariant for non-alternating 11 crossing knots. We will also view obstructions to a knot bounding a Möbius band given by the double branched cover of [Formula: see text] branched over [Formula: see text]. 

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3. Good polynomials for locally recoverable codes: Classification results, asymptotics, and exact formulae

   https://doi.org/10.1142/S0219498825410178  

   Darwish, Mohamed O; Jaindungarwal, Anuvrat; Micheli, Giacomo; Vaghasiya, Divyesh; Zhao, Shujun (March 2025, Journal of Algebra and Its Applications)

   Let [Formula: see text] be a prime power and [Formula: see text]. In this paper we complete the classification of good polynomials of degree [Formula: see text] that achieve the best possible asymptotics (with an explicit error term) for the number of totally split places. Moreover, for degrees up to [Formula: see text], we provide an explicit lower bound and an asymptotic estimate for the number of totally split places of [Formula: see text]. Finally, we prove the general fact that the number [Formula: see text] of [Formula: see text] for which [Formula: see text] splits obeys a linear recurring sequence. For any [Formula: see text], this allows for the computation of [Formula: see text] for large [Formula: see text] by only computing a recurrence sequence over [Formula: see text]. 

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4. Tri-plane diagrams for simple surfaces in S4

   https://doi.org/10.1142/S0218216523500414  

   Allred, Wolfgang; Aragón, Manuel; Dooley, Zack; Goldman, Alexander; Lei, Yucong; Martinez, Isaiah; Meyer, Nicholas; Peters, Devon; Warrander, Scott; Wright, Ana; et al (June 2023, Journal of Knot Theory and Its Ramifications)

   Meier and Zupan proved that an orientable surface [Formula: see text] in [Formula: see text] admits a tri-plane diagram with zero crossings if and only if [Formula: see text] is unknotted, so that the crossing number of [Formula: see text] is zero. We determine the minimal crossing numbers of nonorientable unknotted surfaces in [Formula: see text], proving that [Formula: see text], where [Formula: see text] denotes the connected sum of [Formula: see text] unknotted projective planes with normal Euler number [Formula: see text] and [Formula: see text] unknotted projective planes with normal Euler number [Formula: see text]. In addition, we convert Yoshikawa’s table of knotted surface ch-diagrams to tri-plane diagrams, finding the minimal bridge number for each surface in the table and providing upper bounds for the crossing numbers. 

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5. Compact quantum metric spaces from free graph algebras

   https://doi.org/10.1142/S0129167X22500732  

   Aguilar, Konrad; Hartglass, Michael; Penneys, David (September 2022, International Journal of Mathematics)

   Starting with a vertex-weighted pointed graph [Formula: see text], we form the free loop algebra [Formula: see text] defined in Hartglass–Penneys’ article on canonical [Formula: see text]-algebras associated to a planar algebra. Under mild conditions, [Formula: see text] is a non-nuclear simple [Formula: see text]-algebra with unique tracial state. There is a canonical polynomial subalgebra [Formula: see text] together with a Dirac number operator [Formula: see text] such that [Formula: see text] is a spectral triple. We prove the Haagerup-type bound of Ozawa–Rieffel to verify [Formula: see text] yields a compact quantum metric space in the sense of Rieffel. We give a weighted analog of Benjamini–Schramm convergence for vertex-weighted pointed graphs. As our [Formula: see text]-algebras are non-nuclear, we adjust the Lip-norm coming from [Formula: see text] to utilize the finite dimensional filtration of [Formula: see text]. We then prove that convergence of vertex-weighted pointed graphs leads to quantum Gromov–Hausdorff convergence of the associated adjusted compact quantum metric spaces. As an application, we apply our construction to the Guionnet–Jones–Shyakhtenko (GJS) [Formula: see text]-algebra associated to a planar algebra. We conclude that the compact quantum metric spaces coming from the GJS [Formula: see text]-algebras of many infinite families of planar algebras converge in quantum Gromov–Hausdorff distance. 

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