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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. A lower bound on critical points of the electric potential of a knot

   https://doi.org/10.1142/S0218216521500267  

   Lipton, Max (April 2021, Journal of Knot Theory and Its Ramifications)

   null (Ed.)

   Consider a knot [Formula: see text] in [Formula: see text] with charge uniformly distributed on it. From the standpoint of both physics and knot theory, it is natural to try to understand the critical points of the potential and their behavior. We show the number of critical points of the potential is at least [Formula: see text], where [Formula: see text] is the tunnel number, defined as the smallest number of arcs one must add to [Formula: see text] such that its complement is a handlebody. The result is proven using Morse theory and stable manifold theory. 

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4. 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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5. The multi-region index of a knot

   https://doi.org/10.1142/S0218216520500224  

   Goodhill, Sarah; Lowrance, Adam M.; Gonzales, Valeria Munoz; Rattray, Jessica; Zeh, Amelia (April 2020, Journal of Knot Theory and Its Ramifications)

   Using region crossing changes, we define a new invariant called the multi-region index of a knot. We prove that the multi-region index of a knot is bounded from above by twice the crossing number of the knot. In addition, we show that the minimum number of generators of the first homology of the double branched cover of [Formula: see text] over the knot is strictly less than the multi-region index. Our proof of this lower bound uses Goeritz matrices. 

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