Journal of Knot Theory and Its Ramifications

In this paper, we present a systematic method to generate prime knot and prime link minimal triple-point projections, and then classify all classical prime knots and prime links with triple-crossing number at most four. We also extend the table of known knots and links with triple-crossing number equal to five. By introducing a new type of diagrammatic move, we reduce the number of generating moves on triple-crossing diagrams, and derive a minimal generating set of moves connecting triple-crossing diagrams of the same knot.

In this paper, we give an explicit formula for the twisted Alexander polynomial of any torus link and show that it is a locally constant function on the $\mathrm{\text{SL}}(2,ℂ)$-character variety. We also discuss similar things for the higher-dimensional twisted Alexander polynomial and the Reidemeister torsion.

O’Hara’s energies, introduced by Jun O’Hara, were proposed to answer the question what the canonical shape in a given knot type is, and were configured so that the less the energy value of a knot is, the “better” its shape is. The existence and regularity of minimizers has been well studied. In this paper, we calculate the first and second variational formulae of the $(\alpha ,p)$-O’Hara energies and show absolute integrability, uniform boundedness, and continuity properties. Although several authors have already considered the variational formulae of the $(\alpha ,1)$-O’Hara energies, their techniques do not seem to be applicable to the case $p>1$. We obtain the variational formulae in a novel manner by extracting a certain function from the energy density. All of the $(\alpha ,p)$-energies are made from this function, and by analyzing it, we obtain not only the variational formulae but also the estimates in several function spaces.

The category of finite dimensional modules over the quantum superalgebra $U_q𝔰𝔩(2|1)$ is not semi-simple and the quantum dimension of a generic $U_q𝔰𝔩(2|1)$-module vanishes. This vanishing happens for any value of $q$ (even when $q$ is not a root of unity). These properties make it difficult to create a fusion or modular category. Loosely speaking, the standard way to obtain such a category from a quantum group is: (1) specialize $q$ to a root of unity; this forces some modules to have zero quantum dimension, (2) quotient by morphisms of modules with zero quantum dimension, (3) show the resulting category is finite and semi-simple. In this paper, we show an analogous construction works in the context of $U_q𝔰𝔩(2|1)$ by replacing the vanishing quantum dimension with a modified quantum dimension. In particular, we specialize $q$ to a root of unity, quotient by morphisms of modules with zero modified quantum dimension and show the resulting category is generically finite semi-simple. Moreover, we show the categories of this paper are relative $G$-spherical categories. As a consequence, we obtain invariants of 3-manifold with additional structures.

For an unoriented link ${𝒦}$, let $L({𝒦})$ be the ropelength of ${𝒦}$. It is known that in general $L({𝒦})$ is at least of the order $O({(\mathrm{\text{Cr}}({𝒦}))}^{3/4})$, and at most of the order $O(\mathrm{\text{Cr}}({𝒦}){ln}^5(\mathrm{\text{Cr}}({𝒦}))$ where $\mathrm{\text{Cr}}({𝒦})$ is the minimum crossing number of ${𝒦}$. Furthermore, it is known that there exist families of (infinitely many) links with the property $L({𝒦})=O(\mathrm{\text{Cr}}({𝒦}))$. A long standing open conjecture states that if ${𝒦}$ is alternating, then $L({𝒦})$ is at least of the order $O(\mathrm{\text{Cr}}({𝒦}))$. In this paper, we show that the braid index of a link also gives a lower bound of its ropelength. More specifically, we show that there exists a constant $a>0$ such that $L({𝒦})\ge aB({𝒦})$ for any ${𝒦}$, where $B({𝒦})$ is the largest braid index among all braid indices corresponding to all possible orientation assignments of the components of ${𝒦}$ (called the maximum braid index of ${𝒦}$). Consequently, $L({𝒦})\ge O(\mathrm{\text{Cr}}({𝒦}))$ for any link ${𝒦}$ whose maximum braid index is proportional to its crossing number. In the case of alternating links, the maximum braid indices for many of them are proportional to their crossing numbers hence the above conjecture holds for these alternating links.

In this paper, we prove that every Khovanov homology associated to a Frobenius algebra of rank 2 can be modified in such a way as to produce a TQFT on oriented links, that is a monoidal functor from the category of cobordisms of oriented links to the homotopy category of complexes.

We consider 3-manifolds admitting the action of an involution such that its space of orbits is homeomorphic to $S^3.$ Such involutions are called hyperelliptic as the manifolds admitting such an action. We consider finite groups acting on 3-manifolds and containing hyperelliptic involutions whose fixed-point set has $r>2$ components. In particular we prove that a simple group containing such an involution is isomorphic to $PSL(2,q)$ for some odd prime power $q$, or to one of four other small simple groups.

Using region crossing changes, we define a new invariant called the multi-region index of a knot. We prove that the multi-region index of a knot is bounded from above by twice the crossing number of the knot. In addition, we show that the minimum number of generators of the first homology of the double branched cover of $S^3$ over the knot is strictly less than the multi-region index. Our proof of this lower bound uses Goeritz matrices.

In this paper, we define the minimality of a partition for a critical Heegaard surface. The standard minimal genus Heegaard surface of $(\mathrm{\text{a closed orientable surface}})\times S^1$, which is known to be critical, admits a minimal partition. Moreover, we give an example of a critical surface that admits both a minimal partition and a non-minimal partition.