Journal of Knot Theory and Its Ramifications

L. H. Kauffman defined the binary bracket polynomial of a virtual link by introducing binary labelings into the states of a virtual link diagram. We use the invariant by a slight modification, and call it the modified b-polynomial. We prove that if a virtual link K has a period p^l for a prime p and a positive integer l, then the modified b-polynomial Inv_K (A) of K is congruent to Inv_K* (A) modulo p and A^4pl-1 where K* is the mirror image of K. We exhibit examples of virtual links whose periods are completely determined by the invariant.

We develop a diagrammatic formalism for calculating the Alexander polynomial of the closure of a braid as a state-sum. Our main tools are the Markov trace formulas for the HOMFLY-PT polynomial and Young's semi-normal representations of the Iwahori–Hecke algebras of type A.

We give a method to construct new self-adjoint representations of 𝔹_n of finite dimension. In particular, we give a family of irreducible self-adjoint representations of dimension arbitrarily large. Moreover we give sufficient condition for a representation to be constructed with this method.

The singularity set of a generic standard projection to the three space of a closed surface linked in four space, consists of at most three types: double points, triple points or branch points. We say that this generic projection image is p-diagram if it does not contain any triple point. Two p-diagrams of equivalent surface links are called p-equivalent if there exist a finite sequence of local moves, such that each of them is one of the four moves taken from the seven on the well known Roseman list, that connects only p-diagrams. It is natural to ask that whether any of two p-diagrams of equivalent surface links always p-equivalent? We introduce an invariant of p-equivalent diagrams and an example of linked surfaces that answers our question negatively.

We prove an upper bound for the evaluation of all classical SU₂ spin networks conjectured by Garoufalidis and van der Veen. This implies one half of the analogue of the volume conjecture which they proposed for classical spin networks. We are also able to obtain the other half, namely, an exact determination of the spectral radius, for the special class of generalized drum graphs. Our proof uses a version of Feynman diagram calculus which we developed as a tool for the interpretation of the symbolic method of classical invariant theory, in a manner which is rigorous yet true to the spirit of the classical literature.

We investigate the filtered theory corresponding to the universal sl(2) foam cohomology for links, where a, h ∈ ℂ. We show that there is a spectral sequence converging to which is invariant under the Reidemeister moves, and whose E₁ term is isomorphic to Khovanov homology. This spectral sequence can be used to obtain from the foam perspective an analogue of the Rasmussen invariant and a lower bound for the slice genus of a knot.

This article concerns exact results on the minimum number of colors of a Fox coloring over the integers modulo r, of a link with non-null determinant.

Specifically, we prove that whenever the least prime divisor of the determinant of such a link and the modulus r is 2, 3, 5, or 7, then the minimum number of colors is 2, 3, 4, or 4 (respectively) and conversely.

We are thus led to conjecture that for each prime p there exists a unique positive integer, m_p, with the following property. For any link L of non-null determinant and any modulus r such that p is the least prime divisor of the determinant of L and the modulus r, the minimum number of colors of L modulo r is m_p.

For a knot K the cube number is a knot invariant defined to be the smallest n for which there is a cube diagram of size n for K. Examples of knots for which the cube number detects chirality are presented. An example is also presented of two Legendrian knots that, while topologically isotopic, are distinguished as Legendrian knots by cube number.

We define two kinds of invariants of links in closed 3-manifolds, the s-complexity(s ∈ ℕ) and the block number, by considering decompositions of links in closed orientable 3-manifolds by spines. The first one is a generalization of the complexity of links defined by Pervova and Petronio. After providing properties of these invariants, we construct special spines of strongly-cyclic coverings branched over generalized twist knots in lens spaces, including S³ and ℝP³, which provide upper bounds for the invariants.

Let K be a smooth knot of unit thickness embedded in the space with length L(K) and total curvature κ(K). Then where acn(K) is the average crossing number of the embedded knot K and c > 0 is a constant independent of the knot K. This relationship had been conjectured in [G. Buck and J. Simon, Total curvature and packing of knots, Topology Appl.154 (2007) 192–204] where it is shown that the square root power on the curvature is the lowest possible. In the last section we give several examples to illustrate some relationships between the three quantities average crossing number, total curvature and ropelength.

Let H_g be a genus g handlebody and MCG_2n(T_g) be the group of the isotopy classes of orientation preserving homeomorphisms of T_g = ∂H_g, fixing a given set of 2n points. In this paper we study two particular subgroups of MCG_2n(T_g) which generalize Hilden groups defined by Hilden in [Generators for two groups related to the braid groups, Pacific J. Math.59 (1975) 475–486]. As well as Hilden groups are related to plat closures of braids, these generalizations are related to Heegaard splittings of manifolds and to bridge decompositions of links. Connections between these subgroups and motion groups of links in closed 3-manifolds are also provided.