论文标题
晶格管中聚合物的纠缠统计和4个板
Entanglement statistics of polymers in a lattice tube and unknotting of 4-plats
论文作者
论文摘要
结的熵猜想指出，带有结型$ k $的$ n $边缘格子多边形的指数增长率与未开玩笑多边形的指数增长率相同。此外，下一个订单增长遵循$ n $的权力法，指数在打结$ k $的每个主要结中增加一个。我们通过考虑在管$ \ mathbb {t}^*$中考虑结和非切割链接的第一个证明，这是一个$ \ infty \ times 2 \ times 1 $ sublattice of Simple Cubic lattice的Sublattice。我们建立了$ n $ edge多边形数量的上限和下限，与$ \ mathbb {t}^*$中的固定链接类型的渐近数与$ n $ - edge unkots的数量的固定链接类型。对于上界，我们证明可以通过编织插入来无结。对于下边界，我们使用精确传输诊断的信息证明了一个模式定理。这项工作为晶格多边形提供了4个斑点和新组合学结果的新结理论结果。突出显示了与纳米通道中DNA等建模聚合物的连接。
The Knot Entropy Conjecture states that the exponential growth rate of the number of $n$-edge lattice polygons with knot-type $K$ is the same as that for unknot polygons. Moreover, the next order growth follows a power law in $n$ with an exponent that increases by one for each prime knot in the knot decomposition of $K$. We provide the first proof of this conjecture by considering knots and non-split links in tube $\mathbb{T}^*$, an $\infty \times 2\times 1$ sublattice of the simple cubic lattice. We establish upper and lower bounds relating the asymptotics of the number of $n$-edge polygons with fixed link-type in $\mathbb{T}^*$ to that of the number of $n$-edge unknots. For the upper bound, we prove that polygons can be unknotted by braid insertions. For the lower bound, we prove a pattern theorem for unknots using information from exact transfer-matrices. This work provides new knot theory results for 4-plats and new combinatorics results for lattice polygons. Connections to modelling polymers such as DNA in nanochannels are highlighted.
