Journal of Knot Theory and Its Ramifications

We introduce a local move on a link diagram named a region freeze crossing change which is close to a region crossing change, but not the same. We study similarity and difference between region crossing change and region freeze crossing change.

We construct the complete invariant for fused links. It is proved that the set of equivalence classes of $n$-component fused links is in one-to-one correspondence with the set of elements of the abelization $UV{P}_n/UV{P}_n^{\prime }$ up to conjugation by elements from the symmetric group $S_n<UV{B}_n$.

Two-dimensional knots and links are studied in the paper. In this paper, the notion of a parity for 2D knots and links is introduced via techniques similar to the ones used by the second named author in 1D case. By using parity new invariants are constructed and some known invariants are refined.

Quandle 2-cocycles define invariants of classical and virtual knots, and extensions of quandles. We show that the quandle 2-cocycle invariant with respect to a non-trivial $2$-cocycle is constant, or takes some other restricted form, for classical knots when the corresponding extensions satisfy certain algebraic conditions. In particular, if an abelian extension is a conjugation quandle, then the corresponding cocycle invariant is constant. Specific examples are presented from the list of connected quandles of order less than 48. Relations among various quandle epimorphisms involved are also examined.

Olympic systems are collections of small ring polymers whose aggregate properties are largely characterized by the extent (or absence) of topological linking in contrast with the topological entanglement arising from physical movement constraints associated with excluded volume contacts or arising from chemical bonds. First, discussed by de Gennes, they have been of interest ever since due to their particular properties and their occurrence in natural organisms, for example, as intermediates in the replication of circular DNA in the mitochondria of malignant cells or in the kinetoplast DNA networks of trypanosomes. Here, we study systems that have an intrinsic one, two, or three-dimensional character and consist of large collections of ring polymers modeled using periodic boundary conditions. We identify and discuss the evolution of the dimensional character of the large scale topological linking as a function of density. We identify the critical densities at which infinite linked subsystems, the onset of percolation, arise in the periodic boundary condition systems. These provide insight into the nature of entanglement occurring in such course grained models. This entanglement is measured using Gauss linking number, a measure well adapted to such models. We show that, with increasing density, the topological entanglement of these systems increases in complexity, dimension, and probability.

The $(m,n)$-Turk’s head link is presented by the alternating diagram which is obtained from the standard diagram of the $(m,n)$-torus link by crossing changes. In this paper, we show that for any integers $m\ge 1$ and $n=2,3$, the coefficients of $z^i$ in the Conway polynomials of the $(m,n)$- and $(n,m)$-Turk’s head links coincide for $i\equiv 0,1(mod4)$ and differ by sign for $i\equiv 2,3(mod4)$. We conjecture that this property holds for any $n$.

In this paper, dedicated to Prof. Lou Kauffman, we determine the Thurston’s geometry possesed by any Seifert fibered conemanifold structure in a Seifert manifold with orbit space $S^2$ and no more than three exceptional fibers, whose singular set, composed by fibers, has at most three components which can include exceptional or general fibers (the total number of exceptional and singular fibers is less than or equal to three). We also give the method to obtain the holonomy of that structure. We apply these results to three families of Seifert manifolds, namely, spherical, Nil manifolds and manifolds obtained by Dehn surgery on a torus knot $K_{(r,s)}$. As a consequence we generalize to all torus knots the results obtained in [Geometric conemanifolds structures on ${𝕋}_{p/q}$, the result of $p/q$ surgery in the left-handed trefoil knot $𝕋$, J. Knot Theory Ramifications 24(12) (2015), Article ID: 1550057, 38pp., doi: 10.1142/S0218216515500571] for the case of the left handle trefoil knot. We associate a plot to each torus knot for the different geometries, in the spirit of Thurston.

In this paper, we work toward the Homflypt skein module of the lens spaces $L(p,1)$, ${𝒮}(L(p,1))$ using braids. In particular, we establish the connection between ${𝒮}(\mathrm{\text{ST}})$, the Homflypt skein module of the solid torus ST, and ${𝒮}(L(p,1))$ and arrive at an infinite system, whose solution corresponds to the computation of ${𝒮}(L(p,1))$. We start from the Lambropoulou invariant $X$ for knots and links in ST, the universal analog of the Homflypt polynomial in ST, and a new basis, $\mathrm{\Lambda }$, of ${𝒮}(\mathrm{\text{ST}})$ presented in [I. Diamantis and S. Lambropoulou, A new basis for the Homflypt skein module of the solid torus, J. Pure Appl. Algebra 220(2) (2016) 577–605, http://dx.doi.org/10.1016/j.jpaa.2015.06.014, arXiv:1412.3642 [math.GT]]. We show that ${𝒮}(L(p,1))$ is obtained from ${𝒮}(\mathrm{\text{ST}})$ by considering relations coming from the performance of braid band move(s) [bbm] on elements in the basis $\mathrm{\Lambda }$, where the bbm are performed on any moving strand of each element in $\mathrm{\Lambda }$. We do that by proving that the system of equations obtained from diagrams in ST by performing bbm on any moving strand is equivalent to the system obtained if we only consider elements in the basic set $\mathrm{\Lambda }$. The importance of our approach is that it can shed light on the problem of computing skein modules of arbitrary c.c.o. $3$-manifolds, since any $3$-manifold can be obtained by surgery on $S^3$ along unknotted closed curves. The main difficulty of the problem lies in selecting from the infinitum of band moves some basic ones and solving the infinite system of equations.