Kyungpook Mathematical Journal

  • 제64권3호
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  • Pages.511-518
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  • 2024
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  • 1225-6951(pISSN)
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  • 0454-8124(eISSN)
경북대학교 자연과학대학 수학과 (Department of Mathematics, Kyungpook National University)
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2n-Moves and the Γ-Polynomial for Knots

  • Hideo Takioka (School of Mechanical Engineering, College of Science and Engineering, Kanazawa University)
  • 투고 : 2020.08.22
  • 심사 : 2021.07.12
  • 발행 : 2024.09.30
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초록

A 2n-move is a local change for knots and links which changes 2n-half twists to 0-half twists or vice versa for a natural number n. In 1979, Yasutaka Nakanishi conjectured that the 4-move is an unknotting operation. This is still an open problem. It is known that the Γ-polynomial is an invariant for oriented links which is the common zeroth coefficient polynomial of the HOMFLYPT and Kauffman polynomials. In this paper, we show that the 4k-move is not an unknotting operation for any integer k(≥ 2) by using the Γ-polynomial, and if Γ(K; -1) ≡ 9 (mod 16) then the knot K cannot be deformed into the unknot by a single 4-move. Moreover, we give a one-to-one correspondence between the value Γ(K; -1) (mod 16) and the pair (a2(K), a4(K)) (mod 2) of the second and fourth coefficients of the Alexander-Conway polynomial for a knot K.

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과제정보

This work was supported by JSPS KAKENHI Grant Number JP22K13911.

참고문헌

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