Journal of Knot Theory and Its Ramifications

We construct a linear representation of the CGW algebra of type D_n. This representation has degree n² - n, the number of positive roots of a root system of type D_n. We show that the representation is generically irreducible, but that when the parameters of the algebra are related in a certain way, it becomes reducible. As a representation of the Artin group of type D_n, this representation is equivalent to the faithful linear representation of Cohen–Wales. We give a reducibility criterion for this representation as well as a conjecture on the semisimplicity of the CGW algebra of type D_n. Our proof is computer-assisted using Mathematica.

In this paper, we study the colorability of link diagrams by the Alexander quandles. We show that if the reduced Alexander polynomial Δ_L(t) is vanishing, then L admits a non-trivial coloring by any non-trivial Alexander quandle Q, and that if Δ_L(t) = 1, then L admits only the trivial coloring by any Alexander quandle Q, also show that if Δ_L(t) ≠ 0, 1, then L admits a non-trivial coloring by the Alexander quandle Λ/(Δ_L(t)).

Every virtual knot has a pair of groups called the upper and lower groups of the knot. In this paper we treat two topics on those groups: We first give a sufficient condition for a pair of groups which are realized by a certain virtual knot as the upper and lower groups. Secondly we give a necessary condition for a virtual knot to be almost classical in terms of the first elementary ideals of the groups.

We give the explicit formula of the spin-refined Reshetikhin–Turaev SU(2) invariants of lens spaces. Using this result, we also give the formula of the spin-refined Turaev–Viro SU(2) invariants of lens spaces.

A local move called a C_n-move is closely related to Vassiliev invariants. A C_n-distance between two knots K and L, denoted by d_Cn(K, L), is the minimum number of times of C_n-moves needed to transform K into L. Let p and q be natural numbers with p > q ≥ 1. In this paper, we show that for any pair of knots K₁ and K₂ with d_Cn(K₁, K₂) = p and for any given natural number m, there exist infinitely many knots J_j(j = 1, 2, …) such that d_Cn(K₁, J_j) = q and d_Cn(J_j, K₂) = p - q, and they have the same Vassiliev invariants of order less than or equal to m. In the case of n = 1 or 2, the knots J_j(j = 1, 2, …) satisfy more conditions.

This paper consists of three parts.

First, we generalize the Jaeger Formula to express the Kauffman–Vogel graph polynomial as a state sum of the Murakami–Ohtsuki–Yamada graph polynomial.

Then, we demonstrate that reversing the orientation and the color of a MOY graph along a simple circuit does not change the 𝔰𝔩(N) Murakami–Ohtsuki–Yamada polynomial or the 𝔰𝔩(N) homology of this MOY graph. In fact, reversing the orientation and the color of a component of a colored link only changes the 𝔰𝔩(N) homology by an overall grading shift.

Finally, as an application of the first two parts, we prove that the 𝔰𝔬(6) Kauffman polynomial is equal to the 2-colored 𝔰𝔩(4) Reshetikhin–Turaev link polynomial, which implies that the 2-colored 𝔰𝔩(4) link homology categorifies the 𝔰𝔬(6) Kauffman polynomial.

Arnold introduced invariants J⁺, J^- and St for generic planar curves. It is known that both J⁺/2 + St and J^-/2 + St are invariants for generic spherical curves. Applying these invariants to underlying curves of knot diagrams, we can obtain lower bounds for the number of Reidemeister moves required for unknotting. J^- /2 + St works well to count the minimum number of unmatched RII moves. However, it works only up to a factor of two for RI moves. Let w denote the writhe for a knot diagram. We show that J^-/2 + St ± w/2 also gives sharp counts for the number of required RI moves, and demonstrate that it gives a precise estimate for a certain family of diagrams of the unknot with the underlying curve r = 2 + cos(nθ/(n + 1)), (0 ≤ θ ≤ 2(n + 1)π).

We introduce a polynomial for plane graphs, which is proposed to equal the Dubrovnik polynomial of the corresponding alternating link diagrams via the medial construction. Then using this polynomial we define another polynomial for doubly edge-weighted plane graphs, which has a natural connection with Dubrovnik polynomial of links formed from plane graphs by edge-tangle replacements. In the final, via this connection we give an explanation for a result due to Lipson and compute the Dubrovnik polynomials of classical pretzel links.

In this paper we constructed an isomorphic group of binary matrices to a finite symmetric group. Our method is based on the inversion of permutations. Using this embedding we find an algorithm for writing down a standard braid word representation for each positive permutation braid. Also an algorithm for writing basis of Hecke algebra H_n+1 from such basis of H_n is given.

The fundamental biquandle is an invariant of an oriented surface link, which is defined by a presentation obtained from a surface diagram of the surface link: The generating set consists of labels of the semi-sheets and the relator set consists of relations defined at the double point curves. Any surface link can be presented by a link diagram with some markers which is called a ch-diagram. Using this fact, we give a method for calculating the fundamental biquandle of a surface link from its ch-diagram directly.

It is known that the arc index of alternating knots is the minimal crossing number plus two and the arc index of prime nonalternating knots is less than or equal to the minimal crossing number. We study some cases when the arc index is strictly less than the minimal crossing number. We also give minimal grid diagrams of some prime nonalternating knots with 13 crossings and 14 crossings whose arc index is the minimal crossing number minus one.

We derive symmetries and adjunction inequalities of the knot Floer homology groups which appear to be especially interesting for homologically essential knots. Furthermore, we obtain an adjunction inequality for cobordism maps in knot Floer homologies. We demonstrate the adjunction inequalities and symmetries in explicit calculations which recover some of the main results from [E. Eftekhary, Longitude Floer homology and the Whitehead double, Algebr. Geom. Topol. 5 (2005) 1389–1418] on longitude Floer homology and also give rise to vanishing results on knot Floer homologies.