Symmetric Union Diagrams and Refined Spin Models
Symmetric Union Diagrams and Refined Spin Models
复制标题
对称联合图和精致的自旋模型
DOI：
10.4153/s0008439518000115
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发表时间：
2018
期刊：
影响因子：
--
通讯作者：
P. Lisca
中科院分区：
文献类型：
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作者：
Carlo Collari;P. Lisca
摘要一个类似于切片丝带猜想的开放性问题是，是否每个丝带结都可以表示为一个对称并集。在这个基本的存在性问题之后，是这样的表示的唯一性问题。Eschermann和Lamm通过引入对称并图之间的对称等价的概念来研究后一个问题，并表明可以使用琼斯多项式的改进版本来检测非等价图。我们证明了每个拓扑自旋模型产生了许多有效的对称等价不变量，这些不变量可以用来区分无穷多个Reidemeister等价但对称不等价的对称并图。我们还表明，这样的不变量是不等价的精制琼斯多项式，我们用它们来提供一个部分的答案，一个问题留下了开放的Zuermann和Lamm。
Abstract An open question akin to the slice-ribbon conjecture asks whether every ribbon knot can be represented as a symmetric union. Next to this basic existence question sits the question of uniqueness of such representations. Eisermann and Lamm investigated the latter question by introducing a notion of symmetric equivalence among symmetric union diagrams and showing that non-equivalent diagrams can be detected using a refined version of the Jones polynomial. We prove that every topological spin model gives rise to many effective invariants of symmetric equivalence, which can be used to distinguish infinitely many Reidemeister equivalent but symmetrically non-equivalent symmetric union diagrams. We also show that such invariants are not equivalent to the refined Jones polynomial and we use them to provide a partial answer to a question left open by Eisermann and Lamm.