• 제목/요약/키워드: unknotting Number

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On Minimal Unknotting Crossing Data for Closed Toric Braids

  • Siwach, Vikash;Prabhakar, Madeti
    • Kyungpook Mathematical Journal
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    • 제57권2호
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    • pp.331-360
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    • 2017
  • Unknotting numbers for torus knots and links are well known. In this paper, we present a new approach to determine the position of unknotting number crossing changes in a toric braid such that the closure of the resultant braid is equivalent to the trivial knot or link. Further we give unknotting numbers of more than 600 knots.

Polynomial Unknotting and Singularity Index

  • Mishra, Rama
    • Kyungpook Mathematical Journal
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    • 제54권2호
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    • pp.271-292
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    • 2014
  • We introduce a new method to transform a knot diagram into a diagram of an unknot using a polynomial representation of the knot. Here the unknotting sequence of a knot diagram with least number of crossing changes can be realized by a family of polynomial maps. The number of singular knots in this family is defined to be the singularity index of the diagram. We show that the singularity index of a diagram is always less than or equal to its unknotting number.

Forbidden Detour Number on Virtual Knot

  • Yoshiike, Shun;Ichihara, Kazuhiro
    • Kyungpook Mathematical Journal
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    • 제61권1호
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    • pp.205-212
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    • 2021
  • We show that the forbidden detour move, essentially introduced by Kanenobu and Nelson, is an unknotting operation for virtual knots. Then we define the forbidden detour number of a virtual knot to be the minimal number of forbidden detour moves necessary to transform a diagram of the virtual knot into the trivial knot diagram. Some upper and lower bounds on the forbidden detour number are given in terms of the minimal number of real crossings or the coefficients of the affine index polynomial of the virtual knot.

ARC SHIFT NUMBER AND REGION ARC SHIFT NUMBER FOR VIRTUAL KNOTS

  • Gill, Amrendra;Kaur, Kirandeep;Madeti, Prabhakar
    • 대한수학회지
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    • 제56권4호
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    • pp.1063-1081
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    • 2019
  • In this paper, we formulate a new local move on virtual knot diagram, called arc shift move. Further, we extend it to another local move called region arc shift defined on a region of a virtual knot diagram. We establish that these arc shift and region arc shift moves are unknotting operations by showing that any virtual knot diagram can be turned into trivial knot using arc shift (region arc shift) moves. Based upon the arc shift move and region arc shift move, we define two virtual knot invariants, arc shift number and region arc shift number respectively.

Local Moves and Gordian Complexes, II

  • Nakanishi, Yasutaka
    • Kyungpook Mathematical Journal
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    • 제47권3호
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    • pp.329-334
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    • 2007
  • By the works of Levine [2] and Rolfsen [5], [6], it is known that a local move called a crossing-change is strongly related to the Alexander invariant. In this note, we will consider to what degree the relationship is strong. Let K be a knot, and $K^{\times}$ the set of knots obtained from a knot K by a single crossing-change. Let MK be the Alexander invariant of a knot K, and MK the set of the Alexander invariants $\{MK\}_{K{\in}\mathcal{K}}$ for a set of knots $\mathcal{K}$. Our main result is the following: If both $K_1$ and $K_2$ are knots with unknotting number one, then $MK_1=MK_2$ implies $MK_1^{\times}=MK_2^{\times}$. On the other hand, there exists a pair of knots $K_1$ and $K_2$ such that $MK_1=MK_2$ and $MK_1^{\times}{\neq}MK_2^{\times}$. In other words, the Gordian complex is not homogeneous with respect to Alexander invariants.

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