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...
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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. |
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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 |