A diagrammatic approach to the AJ Conjecture

The AJ Conjecture relates a quantum invariant, a minimal order recursion for the colored Jones polynomial of a knot (known as the $$\hat{A}$$ A ^ polynomial), with a classical invariant, namely the defining polynomial A of the $${\mathrm {PSL}_2(\mathbb {C})}$$ PSL 2 ( C ) character variety of a knot. More precisely, the AJ Conjecture asserts that the set of irreducible factors of the $$\hat{A}$$ A ^ -polynomial (after we set $$q=1$$ q = 1 , and excluding those of L-degree zero) coincides with those of the A-polynomial. In this paper, we introduce a version of the $$\hat{A}$$ A ^ -polynomial that depends on a planar diagram of a knot (that conjecturally agrees with the $$\hat{A}$$ A ^ -polynomial) and we prove that it satisfies one direction of the AJ Conjecture. Our proof uses the octahedral decomposition of a knot complement obtained from a planar projection of a knot, the R-matrix state sum formula for the colored Jones polynomial, and its certificate.

Planar diagrams for local invariants of graphs in surfaces

In order to apply quantum topology methods to nonplanar graphs, we define a planar diagram category that describes the local topology of embeddings of graphs into surfaces. These virtual graphs are a categorical interpretation of ribbon graphs. We describe an extension of the flow polynomial to virtual graphs, the [Formula: see text]-polynomial, and formulate the [Formula: see text] Penrose polynomial for non-cubic graphs, giving contraction–deletion relations. The [Formula: see text]-polynomial is used to define an extension of the Yamada polynomial to virtual spatial graphs, and with it we obtain a sufficient condition for non-classicality of virtual spatial graphs. We conjecture the existence of local relations for the [Formula: see text]-polynomial at squares of integers.

A tabulation of ribbon knots in tangle form

It is known that there are 21 ribbon knots with 10 crossings or fewer. We show that for every ribbon knot, there exists a tangle that satisfies two properties associated with the knot. First, under a specific closure, the closed tangle is equivalent to its corresponding knot. Second, under a different closure, the closed tangle is equivalent to the unlink. For each of these 21 ribbon knots, we present a 4-strand tangle that satisfies these properties. We provide diagrams of these tangles and also express them in planar diagram notation.

An elementary proof of a non-triviality of the E8subfactor planar algebra

In this paper, we show in a combinatorial way that the 0-box space of the E8subfactor planar algebra is 1-dimensional. In the proof, we improve on Bigelow's relations for the E8subfactor planar algebra and give an efficient algorithm to reduce any planar diagram to the empty diagram.

Links and Planar Diagram Codes

In this paper we formalize a combinatorial object for describing link diagrams called a Planar Diagram Code (PD-Code). PD-codes are used by the KnotTheory Mathematica package developed by Bar-Natan et al. We present the set of PD-codes as a standalone object and discuss its relationship with link diagrams. We give an explicit algorithm for reconstructing a knot diagram on a surface from a PD-code. We also discuss the intrinsic symmetries of PD-codes (i.e. invertibility and chirality). The moves analogous to the Reidemeister moves are also explored, and we show that the given set of PD-codes modulo these combinatorial Reidemeister moves is equivalent to classical link theory.

FROM FRAMED LINKS TO CRYSTALLIZATIONS OF BOUNDED 4-MANIFOLDS

It is well-known that every 3-manifold M3 may be represented by a framed link (L,c), which indicates the Dehn-surgery from [Formula: see text] to M3 = M3(L,c); moreover, M3 is the boundary of a PL 4-manifold M4 = M4(L, c), which is obtained from [Formula: see text] by adding 2-handles along the framed link (L, c). In this paper we study the relationships between the above representations and the representation theory of general PL-manifolds by edge-coloured graphs: in particular, we describe how to construct a 5-coloured graph representing M4=M4(L,c), directly from a planar diagram of (L,c). As a consequence, relations between the combinatorial properties of the link L and both the Heegaard genus of M3=M3(L,c) and the regular genus of M4=M4(L,c) are obtained.