The Quantum 𝔰𝔩(3) Invariants of Cubic Bipartite Planar Graphs

Abstract

Temperley-Lieb algebras have been generalized to sl(3, C) web spaces. Since a cubic bipartite planar graph with suitable directions on edges is a web, the quantum sl (3) invariants naturally extend to all cubic bipartite planar graphs. First we completely classify them as a connected sum of primes webs. We also provide a method to find all prime webs and exhibit all prime webs up to 20 vertices. Using quantum sl(3) invariants, we provide a criterion which determine the symmetry of graphs.

ON THE (n, k)-TH CATALAN NUMBERS

“…Thus, it is natural to ask (n, 3) diagrams may be also geometric realizations of the invariant vectors in the invariant space of V ⊗3n 1 where V 1 is the vector representation of sl(3, C). In fact, it is known that there exists a general method to generate all invariant vectors in the invariant space of V ⊗3n 1 [11] and our expectation is right. However, not all invariant vectors are (n, 3) diagrams.…”

“…Kuperberg generalized Temperley-Lieb algebras, which corresponds to the invariant subspace of a tensor product of the vector representation of sl(2), to web spaces of simple Lie algebras of rank 2, sl (3), sp(4) and G 2 [12]. It has been extensively studied [9][10][11]. In particular, a purpose of the present article is to find a generalization which fits in both directions; Gould's generalization and the web spaces of sl(3).…”

“…together with its convolution formula, sl(3), sp(4) and G 2 [12]. It has been extensively studied [9][10][11]. In particular, a purpose of the present article is to find a generalization which fits in both directions; Gould's generalization and the web spaces of sl(3 is that it is equal to the number of triangulations of (n + 2) polygon P n+2 where all vertices of triangles lie on the vertices of P n+2 .…”

A classification of transitive links and periodic links

“…Murasugi also found a similar relation for the Jones polynomials of L and L [17]. There are various result to decide periodicity of links [9,11,20,24,26,27]. These are all necessary conditions for periodic links using polynomial invariants of links.…”