Journal of Knot Theory and Its Ramifications

To systematically construct invariants of handlebody-links, we give a new presentation of the braided tensor category of handlebody-tangles by generators and relations, and prove that given what we call a quantum-commutative quantum-symmetric algebra A in an arbitrary braided tensor category , there arises a braided tensor functor , which gives rise to a desired invariant. Some properties of the invariants and explicit computational results are shown especially when A is a finite-dimensional unimodular Hopf algebra, which is naturally regarded as a quantum-commutative quantum-symmetric algebra in the braided tensor category of Yetter–Drinfeld modules.

This paper is the first one in the sequence of papers about a simple class of framed 4-graphs; the goal of this paper is to collect some well-known results on planarity and to reformulate them in the language of minors. The goal of the whole sequence is to prove analogs of the Robertson–Seymour–Thomas theorems for framed 4-graphs: namely, we shall prove that many minor-closed properties are classified by finitely many excluded graphs. From many points of view, framed 4-graphs are easier to consider than general graphs; on the other hand, framed 4-graphs are closely related to many problems in graph theory.

Y. Miyazawa introduced a two-variable polynomial invariant of virtual knots in 2006 [Magnetic graphs and an invariant for virtual links, J. Knot Theory Ramifications 15 (2006) 1319–1334] and then generalized it to give a multi-variable one via decorated virtual magnetic graph diagrams in 2008. A. Ishii gave a simple state model for the two-variable Miyazawa polynomial by using pole diagrams in 2008 [A multi-variable polynomial invariant for virtual knots and links, J. Knot Theory Ramifications 17 (2008) 1311–1326]. H. A. Dye and L. H. Kauffman constructed an arrow polynomial of a virtual link in 2009 which is equivalent to the multi-variable Miyazawa polynomial [Virtual crossing number and the arrow polynomial, preprint (2008), arXiv:0810.3858v3, http://front.math.ucdavis.edu.]. We give a bracket model for the multi-variable Miyazawa polynomial via pole diagrams and polar tangles similarly to the Ishii's state model for the two-variable polynomial. By normalizing the bracket polynomial we get the multi-variable Miyazawa polynomial f_K ∈ ℤ[A, A^-1, K₁, K₂, …] of a virtual link K. n-similar knots take the same value for any Vassiliev invariant of degree < n. We show that f_K1 ≡ f_K2 mod (A⁴ - 1)ⁿ if two virtual links K₁ and K₂ are n-similar. Also we give a necessary condition for a virtual link to be periodic by using n-similarity of virtual tangles and the Miyazawa polynomial.

This paper considers *-graphs in which all vertices have degree 4 or 6, and studies the question of calculating the genus of nonorientable surfaces into which such graphs may be embedded. In a previous paper [Embeddings of *-graphs into 2-surfaces, preprint (2012), arXiv:1212.5646] by the authors, the problem of calculating whether a given *-graph in which all vertices have degree 4 or 6 admits a ℤ₂-homologically trivial embedding into a given orientable surface was shown to be equivalent to a problem on matrices. Here we extend those results to nonorientable surfaces. The embeddability condition that we obtain yields quadratic-time algorithms to determine whether a *-graph with all vertices of degree 4 or 6 admits a ℤ₂-homologically trivial embedding into the projective plane or into the Klein bottle.

In the paper [First-order invariants and cohomology of spaces of embeddings of self-intersecting curves in ℝⁿ, Izv. Math. 69(5) (2005) 865–912] on finite type invariants of self-intersecting curves, Vassiliev conjectured a criterion of planarity of framed four-valent graphs, i.e. 4-graphs with an opposite edge structure at each vertex. The conjecture was proved by Manturov [A proof of Vassilievs conjecture on the planarity of singular links, Izv. Math. 69(5) (2005) 169–178]. We give here another proof of Vassiliev's planarity criterion of framed four-valent graphs (and more generally, (even) *-graphs), which is based on Pontryagin–Kuratowski theorem.

The involutory birack counting invariant is an integer-valued invariant of unoriented tangles defined by counting homomorphisms from the fundamental involutory birack of the tangle to a finite involutory birack over a set of framings modulo the birack rank of the labeling birack. In this first of an anticipated series of several papers, we enhance the involutory birack counting invariant with quantum weights, which may be understood as tangle functors of involutory birack-labeled unoriented tangles.

Carrell defined the fundamental biquandle of an oriented surface-link by a presentation obtained from its broken surface diagram, which is an invariant up to isomorphism of the fundamental biquandle. Ashihara gave a method to calculate the fundamental biquandle of an oriented surface-link from its marked graph diagram (ch-diagram). In this paper, we discuss the fundamental Alexander biquandles of oriented surface-links via marked graph diagrams, derived computable invariants and their applications to detect non-invertible oriented surface-links.

An equilateral stick number s₌(K) of a knot K is defined to be the minimal number of sticks required to construct a polygonal knot of K which consists of equal length sticks. Rawdon and Scharein [Upper bounds for equilateral stick numbers, in Physical Knots: Knotting, Linking, and Folding Geometric Objects in ℝ³, Contemporary Mathematics, Vol. 304 (American Mathematical Society, Providence, RI, 2002), pp. 55–76] found upper bounds for the equilateral stick numbers of all prime knots through 10 crossings by using algorithms in the software KnotPlot. In this paper, we find an upper bound on the equilateral stick number of a non-trivial knot K in terms of the minimal crossing number c(K) which is s₌(K) ≤ 2c(K) + 2. Moreover if K is a non-alternating prime knot, then s₌(K) ≤ 2c(K) - 2. Furthermore we find another upper bound on the equilateral stick number for composite knots which is s₌(K₁♯K₂) ≤ 2c(K₁) + 2c(K₂).

Let Len(K) be the minimum length of a knot on the cubic lattice (namely the minimum length necessary to construct the knot in the cubic lattice). This paper provides upper bounds for Len(K) of a nontrivial knot K in terms of its crossing number c(K) as follows:

H. Dye defined the parity mapping for a virtual knot diagram, which is a map from the set of real crossings of the diagram to ℤ. The notion generalizes the parity which is studied extensively by V. Manturov. The mapping induces the ith writhe (i ∈ ℤ\{0}) which is an invariant of the representing virtual knot. She applied the parity mapping to introduce a grade to the Henrich S-invariant for a virtual knot, and showed that the invariants are Vassiliev invariants of degree one. Following it, we define the parity mappings for a virtual link diagram, and define the similar invariants as above for a virtual link by using the parity mappings. We show that some of the invariants are Vassiliev invariants of degree one. We also checked necessary conditions for invertibility and amphicheirality via the invariants.

In this paper, we give a relationship between the virtual Alexander polynomial of a periodic virtual knot and that of its factor knot. As an application, we improve Im–Lee's table of possible periods of 117 virtual knots with classical crossings ≤ 4 in Jeremy Green's table. In particular, we prove that the virtual knots 4.1 and 4.77 have the actual period 2 and no others, and the virtual knot 4.99 has the actual period 2.

A Takasaki quandle (T(G), *) is a quandle under the binary operation * defined by a*b = 2b-a for an abelian group (G, +). In this paper, we will show that if a subquandle X of a Takasaki quandle G is a image of subgroup of G under a quandle automorphism of T(G), then the set {X * g | g ∈ G} is a quandle under the binary operation *′ defined by (X * g) *′ (X * h) = X * (g * h). On the other hand, the quotient structure studied in [On quotients of quandles, J. Knot Theory Ramifications 19(9) (2010) 1145–1156] can be applied to the Takasaki quandles. In this paper, we will review the quotient structure studied in [On quotients of quandles, J. Knot Theory Ramifications 19(9) (2010) 1145–1156], and show that the quotient quandle coincides with the quotient quandle defined by Bunch, Lofgren, Rapp and Yetter in [On quotients of quandles, J. Knot Theory Ramifications 19(9) (2010) 1145–1156] for connected Takasaki quandles.

An Alexander quandle M_t is an abelian group M with a quandle operation a * b = ta + (1 - t)b where t is a group automorphism of the abelian group M. In this paper, we will study the commutativity of an Alexander quandle and introduce the relationship between Alexander quandles M_t and M_1-t determined by group automorphisms t and 1 - t, respectively.

We introduce the alternating tangle decomposition of a diagram of a link L and improve the upper bound of arc index α(L) by using information of the alternating tangle decomposition. Also we get the exact arc index of a class of links by combining the upper bound with Morton and Beltrami's lower bound of the arc index.