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1. 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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2. Slope of orderable Dehn filling of two-bridge knots

   https://doi.org/10.1142/S0218216522500067  

   Gao, Xinghua (January 2022, Journal of Knot Theory and Its Ramifications)

   In this paper, we study the Riley polynomial of double twist knots with higher genus. Using the root of the Riley polynomial, we compute the range of rational slope [Formula: see text] such that [Formula: see text]-filling of the knot complement has left-orderable fundamental group. Further more, we make a conjecture about left-orderable surgery slopes of two-bridge knots. 

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3. 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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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. 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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