Journal of Knot Theory and Its Ramifications

A diquandle is a set equipped with two quandle operations interacting via a kind of distributive laws which come from Reidemeister moves on dichromatic links. This algebraic systems provide coloring invariants for dichromatic links. In this paper, we give explicit constructions of free diquandles and diquandle presentations, and then discuss Tietze transformations for the diquandle presentations. We also introduce the fundamental diquandles for dichromatic links. Particularly, we describe the fundamental diquandles and diquandle counting invariants for knots and links in the solid torus via annulus diagrams. We append the tables of diquandles and dikei’s of orders $\le 5$.

A nontrivial knot is called minimal if its knot group does not surject onto the knot groups of other nontrivial knots. In this paper, we determine the minimality of the rational knots $C(2n+1,2m,2)$ in the Conway notation, where $m\ne 0$ and $n\ne 0,-1$ are integers. When $\left|m\right|\ge 2$, we show that the nonabelian ${\mathrm{\text{SL}}}_2(ℂ)$-character variety of $C(2n+1,2m,2)$ is irreducible and therefore $C(2n+1,2m,2)$ is a minimal knot. The proof of this result is an interesting application of Eisenstein’s irreducibility criterion for polynomials over integral domains.

Using the flow property of the $R$-matrix defining the colored Jones polynomial, we establish a natural bijection between the set of states on the part arc-graph of a link projection and the set of states on a corresponding bichromatic digraph, called arc-graph, as defined by Garoufalidis and Loebl [A non-commutative formula for the colored Jones function, Math. Ann. 336 (2006) 867–900]. We use this to give a new and essentially elementary proof for the knot state-sum formula in [S. Garoufalidis and M. Loebl, A non-commutative formula for the colored Jones function, Math. Ann. 336 (2006) 867–900]. We will show that the state-sum contributions of states on the part arc-graph defined by the universal $R$-matrix of $U_q(\mathrm{\text{sl}}(2,ℂ))$ correspond, under our bijection of sets of states, to the contributions in [S. Garoufalidis and M. Loebl, A non-commutative formula for the colored Jones function, Math. Ann. 336 (2006) 867–900]. This will show that the two state models are in fact not essentially distinct. Our approach will also extend the formula of Garoufalidis and Loebl to links. This requires some additional nontrivial observations concerning the geometry of states on the part arc-graphs. We will discuss in detail the computation of the arc-graph state-sum, in particular for $3$-braid closures.

In this paper, we study the invariants of twisted knots derived from the invariants of virtual knots such as the arc shift number and the odd writhe. We provide a class of twisted knots such that the arc shift number of every member is 1.