Journal of Knot Theory and Its Ramifications最新文献

{"title":"An improvement of the lower bound for the minimum number of link colorings by quandles","authors":"H. Abchir, Soukaina Lamsifer","doi":"10.1142/s021821652350044x","DOIUrl":"https://doi.org/10.1142/s021821652350044x","url":null,"abstract":"We improve the lower bound for the minimum number of colors for linear Alexander quandle colorings of a knot given in Theorem 1.2 of Colorings beyond Fox: The other linear Alexander quandles (Linear Algebra and its Applications, Vol. 548, 2018). We express this lower bound in terms of the degree k of the reduced Alexander polynomial of the considered knot. We show that it is exactly k + 1 for L-space knots. Then we apply these results to torus knots and Pretzel knots P(-2,3,2l + 1), l>=0. We note that this lower bound can be attained for some particular knots. Furthermore, we show that Theorem 1.2 quoted above can be extended to links with more that one component.","PeriodicalId":54790,"journal":{"name":"Journal of Knot Theory and Its Ramifications","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47790488","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":"Embedding Alexander quandles into groups","authors":"Toshiyuki Akita","doi":"10.1142/s0218216523500116","DOIUrl":"https://doi.org/10.1142/s0218216523500116","url":null,"abstract":"For any twisted conjugate quandle Q, and in particular any Alexander quandle, there exists a group G such that Q is embedded into the conjugation quandle of G. 1. EMBEDDABLE QUANDLES A non-empty set Q equipped with a binary operation Q×Q → Q, (x,y) 7→ x∗ y is called a quandle if it satisfies the following three axioms: (1) x∗ x = x (x ∈ Q), (2) (x∗ y)∗ z = (x∗ z)∗ (y∗ z) (x,y,z ∈ Q), (3) For all x ∈ Q, the map Sx : Q → Q defined by y 7→ y∗ x is bijective. Quandles were introduced independently by Joyce [7] and Matveev [9]. Since then, quandles have been important objects in the study of knots and links, set-theoretical solutions of the Yang-Baxter equation, Hopf algebras and many others. We refer to Nosaka [10] for further details of quandles. A map f : Q → Q′ of quandles is called a quandle homomorphism if it satisfies f (x ∗ y) = f (x) ∗ f (y) (x,y ∈ Q). Given a group G, the set G equipped with a quandle operation h ∗ g ≔ g−1hg is called the conjugation quandle of G and is denoted by Conj(G). A quandle Q is called embeddable if there exists a group G and an injective quandle homomorphism Q → Conj(G). Not all quandles are embeddable (see the bottom of §2). In their paper [2], Bardakov-Dey-Singh proposed the question “For which quandles X does there exists a group G such that X embeds in the conjugation quandle Conj(G)?” [2, Question 3.1], and proved that Alexander quandles associated with fixed-point free involutions are embeddable [2, Proposition 3.2]. The following is a list of embeddable quandles of which the author is aware: (1) free quandles and free n-quandles (Joyce [7, Theorem 4.1 and Corollary 10.3]), (2) commutative quandles, latin quandles and simple quandles (Bardakov-Nasybullov [3, §5]), (3) core quandles (Bergman [4, (6.5)]), and (4) generalized Alexander quandles associated with fixed-point free automorphisms (Dhanwani-Raundal-Singh [5, Proposition 3.12]). In this short note, we will show that twisted conjugation quandles, which include all Alexander quandles, are embeddable, thereby generalize the aforementioned result of Bardakov-Dey-Singh. 2020 Mathematics Subject Classification. Primary 20N02; Secondary 08A05, 57K10.","PeriodicalId":54790,"journal":{"name":"Journal of Knot Theory and Its Ramifications","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41734436","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":"Self-Dual Maps III: Projective Links","authors":"L. Montejano, J. R. Ramirez Alfonsin, Iván Rasskin","doi":"10.1142/s0218216523500669","DOIUrl":"https://doi.org/10.1142/s0218216523500669","url":null,"abstract":"In this paper, we present necessary and sufficient combinatorial conditions for a link to be projective, that is, a link in $RP^3$. This characterization is closely related to the notions of antipodally self-dual and antipodally symmetric maps. We also discuss the notion of symmetric cycle, an interesting issue arising in projective links leading us to an easy condition to prevent a projective link to be alternating.","PeriodicalId":54790,"journal":{"name":"Journal of Knot Theory and Its Ramifications","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47294975","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":"Δ-Tribrackets and Link Homotopy","authors":"E. Chavez, Sam Nelson","doi":"10.1142/s0218216522500973","DOIUrl":"https://doi.org/10.1142/s0218216522500973","url":null,"abstract":"","PeriodicalId":54790,"journal":{"name":"Journal of Knot Theory and Its Ramifications","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48232965","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":"On closed incompressible meridionally incompressible surfaces in knot and link complements","authors":"Wei Lin","doi":"10.1142/s0218216522500705","DOIUrl":"https://doi.org/10.1142/s0218216522500705","url":null,"abstract":"In this paper, we give a necessary condition for detecting (possibly punctured) closed incompressible meridionally incompressible surfaces in knot or link complements. This condition provides us a method to determine whether an arbitrary link is split or non-split based on the link diagram. We also prove that, up to isotopy, there only exist finitely many such surfaces in non-split prime almost alternating link complements. Finally, we demonstrate an application of the necessary condition by showing elementary by-hand proofs that some certain knots are small knots.","PeriodicalId":54790,"journal":{"name":"Journal of Knot Theory and Its Ramifications","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45199082","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":"HOMFLYPT homology over ℤ2 detects unlinks","authors":"Hao Wu","doi":"10.1142/s0218216522500729","DOIUrl":"https://doi.org/10.1142/s0218216522500729","url":null,"abstract":"We apply the Rasmussen spectral sequence to prove that the [Formula: see text]-graded vector space structure of the HOMFLYPT homology over [Formula: see text] detects unlinks. Our proof relies on a theorem of Batson and Seed stating that the [Formula: see text]-graded vector space structure of the Khovanov homology over [Formula: see text] detects unlinks.","PeriodicalId":54790,"journal":{"name":"Journal of Knot Theory and Its Ramifications","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42747114","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":"Petal Number of Torus Knots Using Superbridge Indices","authors":"Hyoungjun Kim, Sungjong No, Hyungkee Yoo","doi":"10.1142/s0218216522500961","DOIUrl":"https://doi.org/10.1142/s0218216522500961","url":null,"abstract":"A petal projection of a knot $K$ is a projection of a knot which consists of a single multi-crossing and non-nested loops. Since a petal projection gives a sequence of natural numbers for a given knot, the petal projection is a useful model to study knot theory. It is known that every knot has a petal projection. A petal number $p(K)$ is the minimum number of loops required to represent the knot $K$ as a petal projection. In this paper, we find the relation between a superbridge index and a petal number of an arbitrary knot. By using this relation, we find the petal number of $T_{r,s}$ as follows; $$p(T_{r,s})=2s-1$$ when $1<r<s$ and $r equiv 1 mod s-r$. Furthermore, we also find the upper bound of the petal number of $T_{r,s}$ as follows; $$p(T_{r,s})leq2s- 2Biglfloor frac{s}{r} Bigrfloor +1$$ when $s equiv pm 1 mod r$.","PeriodicalId":54790,"journal":{"name":"Journal of Knot Theory and Its Ramifications","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45758582","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":"Braids for unoriented pretzel links","authors":"A. Del Pozo Manglano, P. Manchon","doi":"10.1142/s0218216522500894","DOIUrl":"https://doi.org/10.1142/s0218216522500894","url":null,"abstract":"","PeriodicalId":54790,"journal":{"name":"Journal of Knot Theory and Its Ramifications","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45789753","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":"Almost all 2-bridge knots are curly","authors":"C. Ernst, G. Harrison","doi":"10.1142/s0218216522500845","DOIUrl":"https://doi.org/10.1142/s0218216522500845","url":null,"abstract":"","PeriodicalId":54790,"journal":{"name":"Journal of Knot Theory and Its Ramifications","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44959497","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":"Derivatives of jones polynomials detect delta moves in virtual knots","authors":"Victoria Furlow, Sandy Ganzell, Madison Robinson","doi":"10.1142/s0218216522500821","DOIUrl":"https://doi.org/10.1142/s0218216522500821","url":null,"abstract":"For classical knots, ∆-moves and crossing changes can both unknot any knot. But many virtual knots cannot be unknotted with these moves. Moreover, there are many virtual knots that can be unknotted by crossing changes but cannot be unknotted by ∆-moves. We show that the derivative of the Jones polynomial can detect whether a virtual knot can be unknotted by ∆-moves.","PeriodicalId":54790,"journal":{"name":"Journal of Knot Theory and Its Ramifications","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42373793","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}