论文标题
缠结方程，琼斯的猜想，缠结补充中的表面斜率和Q变形的理性
Tangle Equations, the Jones conjecture, slopes of surfaces in tangle complements, and q-deformed rationals
论文作者
论文摘要
我们研究了$ 2 $ tangle方程的系统，这些方程在分析DNA链的酶作用中起着重要作用。 我们表明，每个框架缠结方程式系统最多都有一个框架有理解决方案。此外，我们表明琼斯未开关的猜想意味着，如果缠结方程系统具有合理的解决方案，那么该解决方案在所有$ 2 $ tangles中都是唯一的。这一结果可能打开了琼斯无结论的纯粹拓扑调用的门。 我们介绍了Kauffman支架比率$ \ {t \} _ q \ in \ mathbb q（q）$ t $ t $ in \ mathbb q（q）$，我们推测，对于$ q = 1 $，它是$ d^3-t t $ d^3-t t $的$ q = 1 $。我们证明了代数$ t $的猜想。我们还证明，对于有理$ t $，括号$ \ {t \} _ q $与Morier-Genoud-ovSienko的$ q $ - 理性重合。 此外，我们将缠结方程式系统与整容手术的猜想和刺激性交叉猜想相关联。
We study systems of $2$-tangle equations which play an important role in the analysis of enzyme actions on DNA strands. We show that every system of framed tangle equations has at most one framed rational solution. Furthermore, we show that the Jones Unknot conjecture implies that if a system of tangle equations has a rational solution then that solution is unique among all $2$-tangles. This result potentially opens a door to a purely topological disproof of the Jones Unknot conjecture. We introduce the notion of the Kauffman bracket ratio $\{T\}_q\in \mathbb Q(q)$ of any $2$-tangle $T$ and we conjecture that for $q=1$ it is the slope of meridionally incompressible surfaces in $D^3-T$. We prove that conjecture for algebraic $T$. We also prove that for rational $T$, the brackets $\{T\}_q$ coincide with the $q$-rationals of Morier-Genoud-Ovsienko. Additionally, we relate systems of tangle equations to the Cosmetic Surgery Conjecture and the Nugatory Crossing Conjecture.
