Scaling Conjecture Regarding the Number of Unknots among Polygons of <i>N</i>≫1 Edges
The conjecture is made based on a plausible, but not rigorous argument, suggesting that the unknot probability for a randomly generated self-avoiding polygon of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2021-08-01
|
| Series: | Physics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2624-8174/3/3/39 |
| _version_ | 1797517603490496512 |
|---|---|
| author | Alexander Y. Grosberg |
| author_facet | Alexander Y. Grosberg |
| author_sort | Alexander Y. Grosberg |
| collection | DOAJ |
| description | The conjecture is made based on a plausible, but not rigorous argument, suggesting that the unknot probability for a randomly generated self-avoiding polygon of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>≫</mo><mn>1</mn></mrow></semantics></math></inline-formula> edges has only logarithmic, and not power law corrections to the known leading exponential law: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>P</mi><mi>unknot</mi></msub><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow><mo>∼</mo><mo form="prefix">exp</mo><mfenced separators="" open="[" close="]"><mo>−</mo><mi>N</mi><mo>/</mo><msub><mi>N</mi><mn>0</mn></msub><mo>+</mo><mi>o</mi><mrow><mo>(</mo><mo form="prefix">ln</mo><mi>N</mi><mo>)</mo></mrow></mfenced></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mn>0</mn></msub></semantics></math></inline-formula> being referred to as the random knotting length. This conjecture is consistent with the numerical result of 2010 by Baiesi, Orlandini, and Stella. |
| first_indexed | 2024-03-10T07:17:50Z |
| format | Article |
| id | doaj.art-ec23908a27804e83b8352e60a0ed7403 |
| institution | Directory Open Access Journal |
| issn | 2624-8174 |
| language | English |
| last_indexed | 2024-03-10T07:17:50Z |
| publishDate | 2021-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Physics |
| spelling | doaj.art-ec23908a27804e83b8352e60a0ed74032023-11-22T14:51:02ZengMDPI AGPhysics2624-81742021-08-013366466810.3390/physics3030039Scaling Conjecture Regarding the Number of Unknots among Polygons of <i>N</i>≫1 EdgesAlexander Y. Grosberg0Center for Soft Matter Research, Department of Physics, New York University, 726 Broadway, New York, NY 10003, USAThe conjecture is made based on a plausible, but not rigorous argument, suggesting that the unknot probability for a randomly generated self-avoiding polygon of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>≫</mo><mn>1</mn></mrow></semantics></math></inline-formula> edges has only logarithmic, and not power law corrections to the known leading exponential law: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>P</mi><mi>unknot</mi></msub><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow><mo>∼</mo><mo form="prefix">exp</mo><mfenced separators="" open="[" close="]"><mo>−</mo><mi>N</mi><mo>/</mo><msub><mi>N</mi><mn>0</mn></msub><mo>+</mo><mi>o</mi><mrow><mo>(</mo><mo form="prefix">ln</mo><mi>N</mi><mo>)</mo></mrow></mfenced></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mn>0</mn></msub></semantics></math></inline-formula> being referred to as the random knotting length. This conjecture is consistent with the numerical result of 2010 by Baiesi, Orlandini, and Stella.https://www.mdpi.com/2624-8174/3/3/39polymersknotsunknot probability |
| spellingShingle | Alexander Y. Grosberg Scaling Conjecture Regarding the Number of Unknots among Polygons of <i>N</i>≫1 Edges Physics polymers knots unknot probability |
| title | Scaling Conjecture Regarding the Number of Unknots among Polygons of <i>N</i>≫1 Edges |
| title_full | Scaling Conjecture Regarding the Number of Unknots among Polygons of <i>N</i>≫1 Edges |
| title_fullStr | Scaling Conjecture Regarding the Number of Unknots among Polygons of <i>N</i>≫1 Edges |
| title_full_unstemmed | Scaling Conjecture Regarding the Number of Unknots among Polygons of <i>N</i>≫1 Edges |
| title_short | Scaling Conjecture Regarding the Number of Unknots among Polygons of <i>N</i>≫1 Edges |
| title_sort | scaling conjecture regarding the number of unknots among polygons of i n i 1 edges |
| topic | polymers knots unknot probability |
| url | https://www.mdpi.com/2624-8174/3/3/39 |
| work_keys_str_mv | AT alexanderygrosberg scalingconjectureregardingthenumberofunknotsamongpolygonsofini1edges |