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1. 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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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. Width of the Whitehead double of a nontrivial knot

   https://doi.org/10.1142/S0218216519500810  

   Guo, Qilong; Li, Zhenkun (December 2019, Journal of Knot Theory and Its Ramifications)

   In this paper, we prove that [Formula: see text], where [Formula: see text] is the width of a knot and [Formula: see text] is the Whitehead double of a nontrivial knot [Formula: see text]. 

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4. 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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5. 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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