Journal of Knot Theory and Its Ramifications最新文献

{"title":"Presentations of diquandles and diquandle coloring invariants for solid torus knots and links","authors":"Jieon Kim, Sang Youl Lee, Mohd Ibrahim Sheikh","doi":"10.1142/s021821652350102x","DOIUrl":"https://doi.org/10.1142/s021821652350102x","url":null,"abstract":"<p>A diquandle is a set equipped with two quandle operations interacting via a kind of distributive laws which come from Reidemeister moves on dichromatic links. This algebraic systems provide coloring invariants for dichromatic links. In this paper, we give explicit constructions of free diquandles and diquandle presentations, and then discuss Tietze transformations for the diquandle presentations. We also introduce the fundamental diquandles for dichromatic links. Particularly, we describe the fundamental diquandles and diquandle counting invariants for knots and links in the solid torus via annulus diagrams. We append the tables of diquandles and dikei’s of orders <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mo>≤</mo><mn>5</mn></math></span><span></span>.</p>","PeriodicalId":54790,"journal":{"name":"Journal of Knot Theory and Its Ramifications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140167232","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Bounds in simple hexagonal lattice and classification of 11-stick knots","authors":"Yueheng Bao, Ari Benveniste, Marion Campisi, Nicholas Cazet, Ansel Goh, Jiantong Liu, Ethan Sherman","doi":"10.1142/s0218216523500979","DOIUrl":"https://doi.org/10.1142/s0218216523500979","url":null,"abstract":"<p>The <i>stick number</i> and the <i>edge length</i> of a knot type in the simple hexagonal lattice (sh-lattice) are the minimal numbers of sticks and edges required, respectively, to construct a knot of the given type in sh-lattice. By introducing a linear transformation between lattices, we prove that for any given knot both values in the sh-lattice are strictly less than the values in the cubic lattice. Finally, we show that the only non-trivial <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mn>1</mn><mn>1</mn></math></span><span></span>-stick knots in the sh-lattice are the trefoil knot (<span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mn>3</mn></mrow><mrow><mn>1</mn></mrow></msub></math></span><span></span>) and the figure-eight knot (<span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mn>4</mn></mrow><mrow><mn>1</mn></mrow></msub></math></span><span></span>).</p>","PeriodicalId":54790,"journal":{"name":"Journal of Knot Theory and Its Ramifications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140147201","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Representations of flat virtual braids which do not preserve the forbidden relations","authors":"Valeriy Bardakov, Bogdan Chuzhinov, Ivan Emel’yanenkov, Maxim Ivanov, Elizaveta Markhinina, Timur Nasybullov, Sergey Panov, Nina Singh, Sergey Vasyutkin, Valeriy Yakhin, Andrei Vesnin","doi":"10.1142/s0218216523500931","DOIUrl":"https://doi.org/10.1142/s0218216523500931","url":null,"abstract":"<p>In the paper, we construct a representation <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>𝜃</mi><mo>:</mo><msub><mrow><mstyle><mtext mathvariant=\"normal\">FVB</mtext></mstyle></mrow><mrow><mi>n</mi></mrow></msub><mo>→</mo><mstyle><mtext mathvariant=\"normal\">Aut</mtext></mstyle><mo stretchy=\"false\">(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub><mo stretchy=\"false\">)</mo></math></span><span></span> of the flat virtual braid group <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mstyle><mtext mathvariant=\"normal\">FVB</mtext></mstyle></mrow><mrow><mi>n</mi></mrow></msub></math></span><span></span> on <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi></math></span><span></span> strands by automorphisms of the free group <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub></math></span><span></span> with <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mn>2</mn><mi>n</mi></math></span><span></span> generators which does not preserve the forbidden relations in the flat virtual braid group. This representation gives a positive answer to the problem formulated by Bardakov in the list of unsolved problems in virtual knot theory and combinatorial knot theory by Fenn <i>et al</i>.</p><p>Also we find the set of normal generators of the groups <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mstyle><mtext mathvariant=\"normal\">VP</mtext></mstyle></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">∩</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span><span></span> in <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mstyle><mtext mathvariant=\"normal\">VB</mtext></mstyle></mrow><mrow><mi>n</mi></mrow></msub></math></span><span></span>, <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mstyle><mtext mathvariant=\"normal\">FVP</mtext></mstyle></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">∩</mo><msub><mrow><mstyle><mtext mathvariant=\"normal\">FH</mtext></mstyle></mrow><mrow><mi>n</mi></mrow></msub></math></span><span></span> in <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mstyle><mtext mathvariant=\"normal\">FVB</mtext></mstyle></mrow><mrow><mi>n</mi></mrow></msub></math></span><span></span>, <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mstyle><mtext mathvariant=\"normal\">GVP</mtext></mstyle></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">∩</mo><msub><mrow><mstyle><mtext mathvariant=\"normal\">GH</mtext></mstyle></mrow><mrow><mi>n</mi></mrow></msub></math></span><span></span> in <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mstyle><mtext mathvariant=\"normal\">GVB</mtext></mstyle></mrow><mrow><mi>n","PeriodicalId":54790,"journal":{"name":"Journal of Knot Theory and Its Ramifications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140147204","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An invariant of virtual trivalent spatial graphs","authors":"Evan Carr, Nancy Scherich, Sherilyn Tamagawa","doi":"10.1142/s0218216523500876","DOIUrl":"https://doi.org/10.1142/s0218216523500876","url":null,"abstract":"<p>We create an invariant of virtual <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>Y</mi></math></span><span></span>-oriented trivalent spatial graphs using colorings by <i>virtual Niebrzydowski algebras</i>. This paper generalizes the color invariants using <i>virtual tribrackets</i> and <i>Niebrzydowski algebras</i> by Nelson–Pico, and Graves-Nelson-T. We computed all tribrackets, Niebrzydowski algebras and virtual Niebrzydowski algebras of orders 3 and 4, and provide generative code for all data sets.</p>","PeriodicalId":54790,"journal":{"name":"Journal of Knot Theory and Its Ramifications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140147213","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Distinguishing Legendrian Knots in Topological Types 74, 948, 10136 with maximal Thurston-Benequin number","authors":"M. Prasolov, V. Shastin","doi":"10.1142/s0218216523501018","DOIUrl":"https://doi.org/10.1142/s0218216523501018","url":null,"abstract":"","PeriodicalId":54790,"journal":{"name":"Journal of Knot Theory and Its Ramifications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139870817","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Distinguishing Legendrian Knots in Topological Types 74, 948, 10136 with maximal Thurston-Benequin number","authors":"M. Prasolov, V. Shastin","doi":"10.1142/s0218216523501018","DOIUrl":"https://doi.org/10.1142/s0218216523501018","url":null,"abstract":"","PeriodicalId":54790,"journal":{"name":"Journal of Knot Theory and Its Ramifications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139811047","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Modified symmetrized integral in G-coalgebras","authors":"Nathan Geer, Ngoc Phu Ha, Bertrand Patureau-Mirand","doi":"10.1142/s0218216523500827","DOIUrl":"https://doi.org/10.1142/s0218216523500827","url":null,"abstract":"<p>For <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>G</mi></math></span><span></span> a commutative group, we give a purely Hopf <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>G</mi></math></span><span></span>-coalgebra construction of <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>G</mi></math></span><span></span>-colored <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mn>3</mn></math></span><span></span>-manifolds invariants using the notion of modified integral.</p>","PeriodicalId":54790,"journal":{"name":"Journal of Knot Theory and Its Ramifications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140147142","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Answer to the referee report on ``Elementary computation of the Jones polynomials for torus links modulo primes''","authors":"Joonoh Kim, Kyoung-Tark Kim","doi":"10.1142/s0218216523501006","DOIUrl":"https://doi.org/10.1142/s0218216523501006","url":null,"abstract":"","PeriodicalId":54790,"journal":{"name":"Journal of Knot Theory and Its Ramifications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139532418","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A new condition on the Jones polynomial of a fibered positive link","authors":"Lizzie Buchanan","doi":"10.1142/s0218216523500797","DOIUrl":"https://doi.org/10.1142/s0218216523500797","url":null,"abstract":"<p>We give a new upper bound on the maximum degree of the Jones polynomial of a fibered positive link. In particular, we prove that the maximum degree of the Jones polynomial of a fibered positive knot is at most four times the minimum degree. Using this result, we can complete the classification of all knots of crossing number <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mo>≤</mo><mn>1</mn><mn>2</mn></math></span><span></span> as positive or not positive, by showing that the seven remaining knots for which positivity was unknown are not positive. That classification was also done independently at around the same time by Stoimenow.</p>","PeriodicalId":54790,"journal":{"name":"Journal of Knot Theory and Its Ramifications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140147121","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Distinguishing surface-links described by 4-charts with 2 crossings and 8 black vertices","authors":"T. Nagase, A. Shima","doi":"10.1142/s021821652350092x","DOIUrl":"https://doi.org/10.1142/s021821652350092x","url":null,"abstract":"","PeriodicalId":54790,"journal":{"name":"Journal of Knot Theory and Its Ramifications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138998354","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}