| 张恺,杨志青.琼斯多项式和4-变换[J].数学年刊A辑,2026,(1):59~66 |
| 琼斯多项式和4-变换 |
| The Jones Polynomial and 4-Moves |
| Received:September 17, 2025 Revised:February 25, 2026 |
| DOI：10.16205/j.cnki.cama.2026.0005 |
| 中文关键词: 琼斯多项式 4-变换 4-变换等价 |
| 英文关键词:Jones polynomial 4-Move 4-Move equivalent |
| 基金项目:国家自然科学基金项目(No.12301083) |
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| 中文摘要: |
| 作者通过分析一次4-变换对定向链环L的琼斯多项式VL(t)的影响, 证明了若分支数为l的定向链环L1和定向链环L2是4-变换等价的, 则当l是奇数时, VL1(?1)?VL2(?1)=4k, 当l是偶数时, VL1(?1)?VL2(?1)=4ki, 其中k∈Z, i2=?1. 并且, 若K1和K2是4-变换等价的纽结, 则V′K1(?1)?V′K2(?1)=4k. 这些结论为我们判断纽结是否4-变换等价于平凡结提供了必要条件. |
| 英文摘要: |
| In this paper, the authors analyze the effect of a 4-move on the Jones polynomial VL(t) associated with an oriented link L. The authors show that if an oriented link L1 with l components is 4-move equivalent to an oriented link L2, then their Jones polynomials satisfy VL1(?1)?VL2(?1)=4k if l is odd, and VL1(?1)?VL2(?1)=4ki if l is even, where k∈Z, i2=?1. Moreover, if two knots K1 and K2 are 4-move equivalent, then V′K1(?1)?V′K2(?1)=4k. These conclusions provide necessary conditions for determining whether a knot is 4-move equivalent to the trivial knot. |
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