Entanglement statistics of polymers in a lattice tube and unknotting of 4-plats
Fuente:
arXiv
Saved in:
| Main Authors: | , , , , , , |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929680906452992 |
|---|---|
| author | Beaton, Nicholas R. Ishihara, Kai Atapour, Mahshid Eng, Jeremy W. Vazquez, Mariel Shimokawa, Koya Soteros, Christine E. |
| author_facet | Beaton, Nicholas R. Ishihara, Kai Atapour, Mahshid Eng, Jeremy W. Vazquez, Mariel Shimokawa, Koya Soteros, Christine E. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_06186 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Entanglement statistics of polymers in a lattice tube and unknotting of 4-plats Beaton, Nicholas R. Ishihara, Kai Atapour, Mahshid Eng, Jeremy W. Vazquez, Mariel Shimokawa, Koya Soteros, Christine E. Geometric Topology Soft Condensed Matter Statistical Mechanics Combinatorics 05A16, 82D60, 57K10 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. |
| title | Entanglement statistics of polymers in a lattice tube and unknotting of 4-plats |
| topic | Geometric Topology Soft Condensed Matter Statistical Mechanics Combinatorics 05A16, 82D60, 57K10 |
| url | https://arxiv.org/abs/2204.06186 |