Journal of Knot Theory and Its Ramifications

Giving a presentation of the group of a 2-braid virtual knot or link, we consider the groups of three families of 2-braid virtual knots. Each of them has a certain feature; for example, we can show: for any positive integer N, there exists a virtual knot group with an element of order N. It is known that the collection of the virtual knot groups is the same as that of the ribbon T²-knot groups. Using our examples we discuss the relationship among the virtual knot groups and other knot groups such as ribbon S²-knot groups, S²-knot groups, T²-knot groups, and S³-knot groups.

We study the effect of Feynman integration and diagrammatic differential operators on the structure of group-like elements in the algebra generated by colored vertex-oriented uni-trivalent graphs. We provide applications of our results to the study of the LMO invariant, a quantum invariant of manifolds. We also indicate further situations in which our results apply and may prove useful. The enumerative approach that we adopt has a clarity that has enabled us to perceive a number of generalizations.

Johnson and Livingston have characterized peripheral structures in homomorphs of knot groups. We extend their approach to the case of links. The main result is an algebraic characterization of all possible peripheral structures in certain homomorphic images of link groups.

Let K be a knot that has an unknotting tunnel τ. We prove that K admits a strong involution that fixes τ pointwise if and only if K is a two-bridge knot and τ its upper or lower tunnel.

We show that graphs of the form G _* K₂ are intrinsically knotted if and only if G is nonplanar. This can be extended to show that G _* K_5m+1 is intrinsically (m + 2)-linked when G is nonplanar. We also apply this result to classify all complete n-partite graphs with respect to intrinsic knotting and show that this family does not produce any new minor-minimal examples. Finally, we categorize all minor-minimal intrinsically knotted graphs on 8 or fewer vertices.

Let β* be the unoriented Sato–Levine invariant, an invariant of two component links with even linking number. For a three component link L, we will develop a relation between the Milnor number , β*(L′) of certain two component sublinks L′ of L, and β* of the link formed by merging two of the components of L by band sum. Using this relation, we will show that every embedding of K₁₀ contains a two component link with β*(L) ≠ 0.

We study the question of whether a finite quandle specified by a matrix is isomorphic to an Alexander quandle. We first collect some necessary conditions for a finite quandle to be Alexander. We then describe an algorithm for determining finding all possible Alexander presentations of a finite quandle given its matrix and provide a URL for Maple code implementing this algorithm.

A two-variable polynomial invariant of non-oriented doubly periodic structures is proposed. A possible application of this polynomial for the classification of textile structures is suggested.

We introduce a notion of topological quandle. Given a topological quandle Q we associate to every classical link L in ℝ³ an invariant J_Q(L) which is a topological space (defined up to a homeomorphism). The space J_Q(L) can be interpreted as a space of colorings of a diagram of the link L with colors from the quandle Q.