Carmen Caprau, Nicolle González, Matthew Hogancamp, Mikhail Mazin
We introduce a family of generalized Schr"oder polynomials $S_tau(q,t,a)$, indexed by triangular partitions $tau$ and prove that $S_tau(q,t,a)$ agrees with the Poincar'e series of the triply graded Khovanov-Rozansky homology of the Coxeter knot $K_tau$ associated to $tau$. For all integers $m,n,dgeq 1$ with $m,n$ relatively prime, the $(d,mnd+1)$-cable of the torus knot $T(m,n)$ appears as a special case. It is known that these knots are algebraic, and as a result we obtain a proof of the $q=1$ specialization of the Oblomkov-Rasmussen-Shende conjecture for these knots. Finally, we show that our Schr"oder polynomial computes the hook components in the Schur expansion of the symmetric function appearing in the shuffle theorem under any line, thus proving a triangular version of the $(q,t)$-Schr"oder theorem.
{"title":"Khovanov-Rozansky homology of Coxeter knots and Schröder polynomials for paths under any line","authors":"Carmen Caprau, Nicolle González, Matthew Hogancamp, Mikhail Mazin","doi":"arxiv-2407.18123","DOIUrl":"https://doi.org/arxiv-2407.18123","url":null,"abstract":"We introduce a family of generalized Schr\"oder polynomials $S_tau(q,t,a)$,\u0000indexed by triangular partitions $tau$ and prove that $S_tau(q,t,a)$ agrees\u0000with the Poincar'e series of the triply graded Khovanov-Rozansky homology of\u0000the Coxeter knot $K_tau$ associated to $tau$. For all integers $m,n,dgeq 1$\u0000with $m,n$ relatively prime, the $(d,mnd+1)$-cable of the torus knot $T(m,n)$\u0000appears as a special case. It is known that these knots are algebraic, and as a\u0000result we obtain a proof of the $q=1$ specialization of the\u0000Oblomkov-Rasmussen-Shende conjecture for these knots. Finally, we show that our\u0000Schr\"oder polynomial computes the hook components in the Schur expansion of\u0000the symmetric function appearing in the shuffle theorem under any line, thus\u0000proving a triangular version of the $(q,t)$-Schr\"oder theorem.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141777254","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The ${rm SL}_3$-skein algebra $mathscr{S}_{bar{q}}(mathfrak{S})$ of a punctured oriented surface $mathfrak{S}$ is a quantum deformation of the coordinate algebra of the ${rm SL}_3$-character variety of $mathfrak{S}$. When $bar{q}$ is a root of unity, we prove the Unicity Theorem for representations of $mathscr{S}_{bar{q}}(mathfrak{S})$, in particular the existence and uniqueness of a generic irreducible representation. Furthermore, we show that the center of $mathscr{S}_{bar{q}}(frak{S})$ is generated by the peripheral skeins around punctures and the central elements contained in the image of the Frobenius homomorphism for $mathscr{S}_{bar{q}}(frak{S})$, a surface generalization of Frobenius homomorphisms of quantum groups related to ${rm SL}_3$. We compute the rank of $mathscr{S}_{bar{q}}(mathfrak{S})$ over its center, hence the dimension of the generic irreducible representation.
{"title":"The Unicity Theorem and the center of the ${rm SL}_3$-skein algebra","authors":"Hyun Kyu Kim, Zhihao Wang","doi":"arxiv-2407.16812","DOIUrl":"https://doi.org/arxiv-2407.16812","url":null,"abstract":"The ${rm SL}_3$-skein algebra $mathscr{S}_{bar{q}}(mathfrak{S})$ of a\u0000punctured oriented surface $mathfrak{S}$ is a quantum deformation of the\u0000coordinate algebra of the ${rm SL}_3$-character variety of $mathfrak{S}$.\u0000When $bar{q}$ is a root of unity, we prove the Unicity Theorem for\u0000representations of $mathscr{S}_{bar{q}}(mathfrak{S})$, in particular the\u0000existence and uniqueness of a generic irreducible representation. Furthermore,\u0000we show that the center of $mathscr{S}_{bar{q}}(frak{S})$ is generated by\u0000the peripheral skeins around punctures and the central elements contained in\u0000the image of the Frobenius homomorphism for $mathscr{S}_{bar{q}}(frak{S})$,\u0000a surface generalization of Frobenius homomorphisms of quantum groups related\u0000to ${rm SL}_3$. We compute the rank of $mathscr{S}_{bar{q}}(mathfrak{S})$\u0000over its center, hence the dimension of the generic irreducible representation.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"81 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141777253","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Akutsu-Deguchi-Ohtsuki (ADO) invariants are the most studied quantum link invariants coming from a non-semisimple tensor category. We show that, for fibered links in $S^3$, the degree of the ADO invariant is determined by the genus and the top coefficient is a root of unity. More precisely, we prove that the top coefficient is determined by the Hopf invariant of the plane field of $S^3$ associated to the fiber surface. Our proof is based on the genus bounds established in our previous work, together with a theorem of Giroux-Goodman stating that fiber surfaces in the three-sphere can be obtained from a disk by plumbing/deplumbing Hopf bands.
{"title":"Non-semisimple $mathfrak{sl}_2$ quantum invariants of fibred links","authors":"Daniel López Neumann, Roland van der Veen","doi":"arxiv-2407.15561","DOIUrl":"https://doi.org/arxiv-2407.15561","url":null,"abstract":"The Akutsu-Deguchi-Ohtsuki (ADO) invariants are the most studied quantum link\u0000invariants coming from a non-semisimple tensor category. We show that, for\u0000fibered links in $S^3$, the degree of the ADO invariant is determined by the\u0000genus and the top coefficient is a root of unity. More precisely, we prove that\u0000the top coefficient is determined by the Hopf invariant of the plane field of\u0000$S^3$ associated to the fiber surface. Our proof is based on the genus bounds\u0000established in our previous work, together with a theorem of Giroux-Goodman\u0000stating that fiber surfaces in the three-sphere can be obtained from a disk by\u0000plumbing/deplumbing Hopf bands.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"55 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141777255","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In 2023 we obtained a $Q$-polynomial structure for the projective geometry $L_N(q)$. In the present paper, we display a more general $Q$-polynomial structure for $L_N(q)$. Our new $Q$-polynomial structure is defined using a free parameter $varphi$ that takes any positive real value. For $varphi=1$ we recover the original $Q$-polynomial structure. We interpret the new $Q$-polynomial structure using the quantum group $U_{q^{1/2}}(mathfrak{sl}_2)$ in the equitable presentation. We use the new $Q$-polynomial structure to obtain analogs of the four split decompositions that appear in the theory of $Q$-polynomial distance-regular graphs.
{"title":"Projective geometries, $Q$-polynomial structures, and quantum groups","authors":"Paul Terwilliger","doi":"arxiv-2407.14964","DOIUrl":"https://doi.org/arxiv-2407.14964","url":null,"abstract":"In 2023 we obtained a $Q$-polynomial structure for the projective geometry\u0000$L_N(q)$. In the present paper, we display a more general $Q$-polynomial\u0000structure for $L_N(q)$. Our new $Q$-polynomial structure is defined using a\u0000free parameter $varphi$ that takes any positive real value. For $varphi=1$ we\u0000recover the original $Q$-polynomial structure. We interpret the new\u0000$Q$-polynomial structure using the quantum group $U_{q^{1/2}}(mathfrak{sl}_2)$\u0000in the equitable presentation. We use the new $Q$-polynomial structure to\u0000obtain analogs of the four split decompositions that appear in the theory of\u0000$Q$-polynomial distance-regular graphs.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141777256","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we give a constructive proof to show that if there exist a classical linear code C is a subset of F_q^n of dimension k and a classical linear code D is a subset of F_q^k^m of dimension s, where q is a power of a prime number p, then there exists an [[nm, ks, d]]_q quantum stabilizer code with d determined by C and D by identifying the stabilizer group of the code. In the construction, we use a particular type of Butson Hadamard matrices equivalent to multiple Kronecker products of the Fourier matrix of order p. We also consider the same construction of a quantum code for a general normalized Butson Hadamard matrix and search for a condition for the quantum code to be a stabilizer code.
在本文中，我们给出了一个构造性证明，表明如果存在一个经典线性码 C 是维数为 k 的 F_q^n 的子集，一个经典线性码 D 是维数为 s 的 F_q^k^m 的子集，其中 q 是时间数 p 的幂，那么存在一个 [[nm, ks, d]]_q量子稳定器码，其中 d 由 C 和 D 通过识别码的稳定器组决定。在构造中，我们使用了一种特定类型的布特森哈达玛矩阵，它等价于 p 阶傅里叶矩阵的多个克朗克乘积。我们还考虑了一般归一化布特森哈达玛矩阵的量子密码的相同构造，并寻找量子密码成为稳定器密码的条件。
{"title":"A Construction of Quantum Stabilizer Codes from Classical Codes and Butson Hadamard Matrices","authors":"Bulent Sarac, Damla Acar","doi":"arxiv-2407.13527","DOIUrl":"https://doi.org/arxiv-2407.13527","url":null,"abstract":"In this paper, we give a constructive proof to show that if there exist a\u0000classical linear code C is a subset of F_q^n of dimension k and a classical\u0000linear code D is a subset of F_q^k^m of dimension s, where q is a power of a\u0000prime number p, then there exists an [[nm, ks, d]]_q quantum stabilizer code\u0000with d determined by C and D by identifying the stabilizer group of the code.\u0000In the construction, we use a particular type of Butson Hadamard matrices\u0000equivalent to multiple Kronecker products of the Fourier matrix of order p. We\u0000also consider the same construction of a quantum code for a general normalized\u0000Butson Hadamard matrix and search for a condition for the quantum code to be a\u0000stabilizer code.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"62 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141741620","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
As generalizations of inverse semibraces introduced by Catino, Mazzotta and Stefanelli, Miccoli has introduced regular $star$-semibraces under the name of involution semibraces and given a sufficient condition under which the associated map to a regular $star$-semibrace is a set-theoretic solution of the Yang-Baxter equation. From the viewpoint of universal algebra, regular $star$-semibraces are (2,2,1)-type algebras. In this paper we continue to study set-theoretic solutions of the Yang-Baxter equation and regular $star$-semibraces. We first consider several kinds of (2,2,1)-type algebras that induced by regular $star$-semigroups and give some equivalent characterizations of the statement that they form regular $star$-semibraces. Then we give sufficient and necessary conditions under which the associated maps to these (2,2,1)-type algebras are set-theoretic solutions of the Yang-Baxter equation. Finally, as analogues of weak braces defined by Catino, Mazzotta, Miccoli and Stefanelli, we introduce weak $star$-braces in the class of regular $star$-semibraces, describe their algebraic structures and prove that the associated maps to weak $star$-braces are always set-theoretic solutions of the Yang-Baxter equation. The result of the present paper shows that the class of completely regular, orthodox and locally inverse regular $star$-semigroups is a source of possibly new set-theoretic solutions of the Yang-Baxter equation. Our results establish the close connection between the Yang-Baxter equation and the classical structural theory of regular $star$-semigroups.
{"title":"Set-theoretic solutions of the Yang-Baxter equation and regular *-semibraces","authors":"Qianxue Liu, Shoufeng Wang","doi":"arxiv-2407.12533","DOIUrl":"https://doi.org/arxiv-2407.12533","url":null,"abstract":"As generalizations of inverse semibraces introduced by Catino, Mazzotta and\u0000Stefanelli, Miccoli has introduced regular $star$-semibraces under the name of\u0000involution semibraces and given a sufficient condition under which the\u0000associated map to a regular $star$-semibrace is a set-theoretic solution of\u0000the Yang-Baxter equation. From the viewpoint of universal algebra, regular\u0000$star$-semibraces are (2,2,1)-type algebras. In this paper we continue to\u0000study set-theoretic solutions of the Yang-Baxter equation and regular\u0000$star$-semibraces. We first consider several kinds of (2,2,1)-type algebras\u0000that induced by regular $star$-semigroups and give some equivalent\u0000characterizations of the statement that they form regular $star$-semibraces.\u0000Then we give sufficient and necessary conditions under which the associated\u0000maps to these (2,2,1)-type algebras are set-theoretic solutions of the\u0000Yang-Baxter equation. Finally, as analogues of weak braces defined by Catino,\u0000Mazzotta, Miccoli and Stefanelli, we introduce weak $star$-braces in the class\u0000of regular $star$-semibraces, describe their algebraic structures and prove\u0000that the associated maps to weak $star$-braces are always set-theoretic\u0000solutions of the Yang-Baxter equation. The result of the present paper shows\u0000that the class of completely regular, orthodox and locally inverse regular\u0000$star$-semigroups is a source of possibly new set-theoretic solutions of the\u0000Yang-Baxter equation. Our results establish the close connection between the\u0000Yang-Baxter equation and the classical structural theory of regular\u0000$star$-semigroups.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"38 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141741627","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The quantum double $D(G)=Bbb C(G)rtimes Bbb C G$ of a finite group plays an important role in the Kitaev model for quantum computing, as well as in associated TQFT's, as a kind of Poincar'e group. We interpret the known construction of its irreps, which are quasiparticles for the model, in a geometric manner strictly analogous to the Wigner construction for the usual Poincar'e group of $Bbb R^{1,3}$. Irreps are labelled by pairs $(C, pi)$, where $C$ is a conjugacy class in the role of a mass-shell, and $pi$ is a representation of the isotropy group $C_G$ in the role of spin. The geometric picture entails $D^vee(G)to Bbb C(C_G)blacktriangleright!!!!< Bbb C G$ as a quantum homogeneous bundle where the base is $G/C_G$, and $D^vee(G)to Bbb C(G)$ as another homogeneous bundle where the base is the group algebra $Bbb C G$ as noncommutative spacetime. Analysis of the latter leads to a duality whereby the differential calculus and solutions of the wave equation on $Bbb C G$ are governed by irreps and conjugacy classes of $G$ respectively, while the same picture on $Bbb C(G)$ is governed by the reversed data. Quasiparticles as irreps of $D(G)$ also turn out to classify irreducible bicovariant differential structures $Omega^1_{C, pi}$ on $D^vee(G)$ and these in turn correspond to braided-Lie algebras $mathcal{L}_{C, pi}$ in the braided category of $G$-crossed modules, which we call `braided racks' and study. We show under mild assumptions that $U(mathcal{L}_{C,pi})$ quotients to a braided Hopf algebra $B_{C,pi}$ related by transmutation to a coquasitriangular Hopf algebra $H_{C,pi}$.
{"title":"Quantum geometric Wigner construction for $D(G)$ and braided racks","authors":"Shahn Majid, Leo Sean McCormack","doi":"arxiv-2407.11835","DOIUrl":"https://doi.org/arxiv-2407.11835","url":null,"abstract":"The quantum double $D(G)=Bbb C(G)rtimes Bbb C G$ of a finite group plays\u0000an important role in the Kitaev model for quantum computing, as well as in\u0000associated TQFT's, as a kind of Poincar'e group. We interpret the known\u0000construction of its irreps, which are quasiparticles for the model, in a\u0000geometric manner strictly analogous to the Wigner construction for the usual\u0000Poincar'e group of $Bbb R^{1,3}$. Irreps are labelled by pairs $(C, pi)$,\u0000where $C$ is a conjugacy class in the role of a mass-shell, and $pi$ is a\u0000representation of the isotropy group $C_G$ in the role of spin. The geometric\u0000picture entails $D^vee(G)to Bbb C(C_G)blacktriangleright!!!!< Bbb C G$\u0000as a quantum homogeneous bundle where the base is $G/C_G$, and $D^vee(G)to\u0000Bbb C(G)$ as another homogeneous bundle where the base is the group algebra\u0000$Bbb C G$ as noncommutative spacetime. Analysis of the latter leads to a\u0000duality whereby the differential calculus and solutions of the wave equation on\u0000$Bbb C G$ are governed by irreps and conjugacy classes of $G$ respectively,\u0000while the same picture on $Bbb C(G)$ is governed by the reversed data.\u0000Quasiparticles as irreps of $D(G)$ also turn out to classify irreducible\u0000bicovariant differential structures $Omega^1_{C, pi}$ on $D^vee(G)$ and\u0000these in turn correspond to braided-Lie algebras $mathcal{L}_{C, pi}$ in the\u0000braided category of $G$-crossed modules, which we call `braided racks' and\u0000study. We show under mild assumptions that $U(mathcal{L}_{C,pi})$ quotients\u0000to a braided Hopf algebra $B_{C,pi}$ related by transmutation to a\u0000coquasitriangular Hopf algebra $H_{C,pi}$.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141722078","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we study a Kahler structure on finite points. In particular, we study the edge Laplacian of a graph twisted by the Kahler structure introduced in this paper. We also discuss a metric aspect from a twisted holomorphic Dolbeault-Dirac spectral triple and show that the points have a finite diameter with respect to Connes' distance.
{"title":"Twisted edge Laplacians on finite graphs from a Kähler structure","authors":"Soumalya Joardar, Atibur Rahaman","doi":"arxiv-2407.11400","DOIUrl":"https://doi.org/arxiv-2407.11400","url":null,"abstract":"In this paper we study a Kahler structure on finite points. In particular, we\u0000study the edge Laplacian of a graph twisted by the Kahler structure introduced\u0000in this paper. We also discuss a metric aspect from a twisted holomorphic\u0000Dolbeault-Dirac spectral triple and show that the points have a finite diameter\u0000with respect to Connes' distance.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141718012","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the paper, a new algebra ${mathcal A}(n)$, which is generated by an upper triangular generating matrix with triple relations, is introduced. It is shown that there exists an isomorphism between the algebra ${mathcal A}(n)$ and the higher Askey-Wilson algebra ${mathfrak{aw}}(n)$ introduced by Cramp'{e}, Frappat et al. Furthermore, we establish a series of automorphisms of ${mathcal A}(n),$ which satisfy braid group relations and coincide with those in ${mathfrak{aw}}(n).$
{"title":"On the higher-rank Askey-Wilson algebras","authors":"Wanxia Wang, Shilin Yang","doi":"arxiv-2407.10404","DOIUrl":"https://doi.org/arxiv-2407.10404","url":null,"abstract":"In the paper, a new algebra ${mathcal A}(n)$, which is generated by an upper\u0000triangular generating matrix with triple relations, is introduced. It is shown\u0000that there exists an isomorphism between the algebra ${mathcal A}(n)$ and the\u0000higher Askey-Wilson algebra ${mathfrak{aw}}(n)$ introduced by Cramp'{e},\u0000Frappat et al. Furthermore, we establish a series of automorphisms of\u0000${mathcal A}(n),$ which satisfy braid group relations and coincide with those\u0000in ${mathfrak{aw}}(n).$","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141718050","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In "Homfly polynomial via an invariant of colored plane graphs", Murakami, Ohtsuki, and Yamada provide a state-sum description of the level $n$ Jones polynomial of an oriented link in terms of a suitable braided monoidal category whose morphisms are $mathbb{Q}[q,q^{-1}]$-linear combinations of oriented trivalent planar graphs, and give a corresponding description for the HOMFLY-PT polynomial. We extend this construction and express the Khovanov-Rozansky homology of an oriented link in terms of a combinatorially defined category whose morphisms are equivalence classes of formal complexes of (formal direct sums of shifted) oriented trivalent plane graphs. By working combinatorially, one avoids many of the computational difficulties involved in the matrix factorization computations of the original Khovanov-Rozansky formulation: one systematically uses combinatorial relations satisfied by these matrix factorizations to simplify the computation at a level that is easily handled. By using this technique, we are able to provide a computation of the level $n$ Khovanov-Rozansky invariant of the 2-strand braid link with $k$ crossings, for arbitrary $n$ and $k$, confirming and extending previous results and conjectural predictions by Anokhina-Morozov, Beliakova-Putyra-Wehrli, Carqueville-Murfet, Dolotin-Morozov, Gukov-Iqbal-Kozcaz-Vafa, Nizami-Munir-Sohail-Usman, and Rasmussen.
在 "通过彩色平面图的不变量的琼斯多项式"（Homfly polynomial via an invariant of colored plane graphs）一文中，Murakami、Ohtsuki 和 Yamada 用合适的编织一元范畴（其形态是面向三价平面图的$mathbb{Q}[q,q^{-1}]$线性组合）提供了面向链路的 $n$ 级琼斯多项式的状态和描述，并给出了相应的 HOMFLY-PTpolynomial 描述。我们扩展了这一构造，并用一个组合定义的范畴来表达有向链接的霍瓦诺夫-罗赞斯基同调，该范畴的态是有向三价平面图（移位的形式直方和）的形式复数的等价类。通过组合工作，我们避免了原始霍瓦诺夫-罗赞斯基公式的矩阵因式分解计算所涉及的许多计算困难：我们系统地使用这些矩阵因式分解所满足的组合关系，将计算简化到易于处理的水平。通过使用这种技术，我们能够在任意的 $n$ 和 $k$ 条件下，计算具有 $k$ 交叉的双股辫状链接的 $n$Khovanov-Rozansky 层不变式、证实并扩展了阿诺基纳-莫罗佐夫、贝利亚科娃-普蒂拉-韦尔利、卡克维尔-穆尔费特、多洛廷-莫罗佐夫、古科夫-伊克巴尔-科兹卡兹-瓦法、尼扎米-穆尼尔-索海尔-乌斯曼和拉斯穆森之前的结果和猜想预测。
{"title":"Computing the Khovanov homology of 2 strand braid links via generators and relations","authors":"Domenico Fiorenza, Omid Hurson","doi":"arxiv-2407.09785","DOIUrl":"https://doi.org/arxiv-2407.09785","url":null,"abstract":"In \"Homfly polynomial via an invariant of colored plane graphs\", Murakami,\u0000Ohtsuki, and Yamada provide a state-sum description of the level $n$ Jones\u0000polynomial of an oriented link in terms of a suitable braided monoidal category\u0000whose morphisms are $mathbb{Q}[q,q^{-1}]$-linear combinations of oriented\u0000trivalent planar graphs, and give a corresponding description for the HOMFLY-PT\u0000polynomial. We extend this construction and express the Khovanov-Rozansky\u0000homology of an oriented link in terms of a combinatorially defined category\u0000whose morphisms are equivalence classes of formal complexes of (formal direct\u0000sums of shifted) oriented trivalent plane graphs. By working combinatorially,\u0000one avoids many of the computational difficulties involved in the matrix\u0000factorization computations of the original Khovanov-Rozansky formulation: one\u0000systematically uses combinatorial relations satisfied by these matrix\u0000factorizations to simplify the computation at a level that is easily handled.\u0000By using this technique, we are able to provide a computation of the level $n$\u0000Khovanov-Rozansky invariant of the 2-strand braid link with $k$ crossings, for\u0000arbitrary $n$ and $k$, confirming and extending previous results and\u0000conjectural predictions by Anokhina-Morozov, Beliakova-Putyra-Wehrli,\u0000Carqueville-Murfet, Dolotin-Morozov, Gukov-Iqbal-Kozcaz-Vafa,\u0000Nizami-Munir-Sohail-Usman, and Rasmussen.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141718014","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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