Journal of Knot Theory and Its Ramifications

In [F. Zhang, R. Qiu and Y. Zou, The subset of $R^3$ realizing metrics on the curve complex, Topology Appl. 193 (2015) 259–269], they defined a subset ${𝒱}$ of $R^3$ and a metric $d_N$ through each point of ${𝒱}$ for the curve complex ${𝒞}(S)$. For further understanding the curve complex, we concern the whole set $R^3$. With adaption to the flow theory on torus, we prove that for any point of $R^3-{𝒱}$, the $d_N$ is not a metric on ${𝒞}(S)$ or ${{𝒞}}^0(S)$. This means that the ${𝒱}$ is the maximal subset of $R^3$ realizing metrics on the curve complex.

We extend the concepts of trivializing and knotting numbers for knots to spatial graphs and 2-bouquet graphs, in particular. Furthermore, we calculate the trivializing and knotting numbers for projections and pseudodiagrams of 2-bouquet spatial graphs based on the number of precrossings and the placement of the precrossings in the pseudodiagram of the spatial graph.

We introduce defects, with internal gauge symmetries, on a knot and Seifert surface to a knot into the combinatorial construction of finite gauge-group Dijkgraaf–Witten theory. The appropriate initial data for the construction are certain three object categories, with coefficients satisfying a partially degenerate cocycle condition.

According to work of Hartley and Kawauchi in 1979 and 1980, the Conway Polynomial of all negative amphicheiral knots and strongly positive amphicheiral knots factors as $\varphi (z)\varphi (-z)$ for some $\varphi (z)\in \mathbb{Z}[z]$. Moreover, a 2012 example due to Ermotti, Hongler and Weber shows that this is not true for general amphicheiral knots. On the other hand, in 2006, the first author made a conjecture equivalent to saying that the Conway polynomial of all amphicheiral knots splits as $\varphi (z)\varphi (-z)$ in the ring ${\mathbb{Z}}_4[z]$. In this paper, we establish this conjecture for all periodically amphicheiral knots built from braids, where the period preserves the braid structure. We also give counterexamples to conjectures on the leading coefficient of the Conway polynomial of an amphicheiral knot due to Stoimenow.

We investigate the space $C(X)$ of images of linearly embedded finite simplicial complexes in ${ℝ}^n$ isomorphic to a given complex $X$, focusing on two special cases: $X$ is the $(n-2)$-skeleton $K$ of an $n$-simplex, and $X$ is the $(n-2)$-skeleton $L$ of an $(n+1)$-simplex, so $X$ has codimension 2 in ${ℝ}^n$, in both cases. The main result is that for $n>2$, $C(X)$ (for either $X=K,L$) deformation retracts to a subspace homeomorphic to the double mapping cylinder

The Jones polynomial $V_L(t)$ for an oriented link $L$ is a one-variable Laurent polynomial link invariant discovered by Jones. For any integer $n\ge 3$, we show that: (1) the difference of Jones polynomials for two oriented links which are $C_n$-equivalent is divisible by ${(t-1)}^n(t^2+t+1)(t^2+1)$, and (2) there exists a pair of two oriented knots which are $C_n$-equivalent such that the difference of the Jones polynomials for them equals ${(t-1)}^n(t^2+t+1)(t^2+1)$.

We give an explicit basis ${ℬ}$ of the quotient of the Kauffman bracket skein algebra ${𝒮}(\mathrm{\Sigma })$ on a surface $\mathrm{\Sigma }$ by the square of an augmentation ideal. Moreover, we construct an embedding of the mapping class group of a compact connected oriented surface of genus $0$ into the Kauffman bracket skein algebra on the surface completed with respect to a filtration coming from the augmentation ideal.

Knot mosaic theory was introduced by Lomonaco and Kauffman in the paper on ‘Quantum knots and mosaics’ to give a precise and workable definition of quantum knots, intended to represent an actual physical quantum system. A knot $(m,n)$-mosaic is an $m\times n$ matrix whose entries are eleven mosaic tiles, representing a knot or a link by adjoining properly. In this paper, we introduce two variants of knot mosaics: period knot mosaics and toroidal knot mosaics, which are common features in physics and mathematics. We present an algorithm producing the exact enumeration of period knot $(m,n)$-mosaics for any positive integers $m$ and $n$, toroidal knot $(m,n)$-mosaics for co-prime integers $m$ and $n$, and furthermore toroidal knot $(p,p)$-mosaics for a prime number $p$. We also analyze the asymptotics of the growth rates of their cardinality.

Since the Jones polynomial was discovered, the connection between knot theory and quantum physics has been of great interest. Lomonaco and Kauffman introduced the knot mosaic system to give a definition of the quantum knot system that is intended to represent an actual physical quantum system. Recently the authors developed an algorithm producing the exact enumeration of knot mosaics, which uses a recursion formula of state matrices. As a sequel to this research program, we similarly define the (embedded) graph mosaic system by using 16 graph mosaic tiles, representing graph diagrams with vertices of valence 3 and 4. We extend the algorithm to produce the exact number of all graph mosaics. The magnified state matrix that is an extension of the state matrix is mainly used.

A revised proof is given to an assertion on chord diagrams of a ribbon surface-link.

We introduce a new class of quantum enhancements we call biquandle brackets, which are customized skein invariants for biquandle colored links. Quantum enhancements of biquandle counting invariants form a class of knot and link invariants that includes biquandle cocycle invariants and skein invariants such as the HOMFLY-PT polynomial as special cases, providing an explicit unification of these apparently unrelated types of invariants. We provide examples demonstrating that the new invariants are not determined by the biquandle counting invariant, the knot quandle or the knot group.