Journal of Knot Theory and Its Ramifications

We apply the theory of directed topology developed by Grandis [Directed homotopy theory, I. The fundamental category, Cah. Topol. Géom. Différ. Catég. 44 (2003) 281–316; Directed Algebraic Topology: Models of Non-Reversible Worlds, New Mathematical Monographs, Vol. 13 (Cambridge University Press, Cambridge, 2009)] to the study of stratified spaces by describing several ways in which a stratification or a stratification with orientations on the strata can be used to produce a related directed space structure. This description provides a setting for the constructions of state-sum TQFTs with defects of [A. L. Dougherty, H. Park and D. N. Yetter, On 2-dimensional Dikjgraaf-Witten theory with defects, J. Knot Theory Ramifications 25(5) (2016) 1650021, doi:10.1142/S0218216516500218; I. J. Lee and D. N. Yetter, Dijkgraaf–Witten type invariants of Seifert surfaces in 3-manifolds, J. Knot Theory Ramifications 26(5) (2017) 1750026, doi:10.1142/S0218216517500262], which we extend to a similar construction of a Dijkgraaf–Witten type TQFT in the case where the defects (lower-dimensional strata) are not sources or targets, but sources on one side and targets on the other, according to an orientation convention.

This paper gives new upper bounds on the stick numbers of the knots $9_{18}$, $10_{18}$, $10_{58}$, $10_{66}$, $10_{68}$, $10_{80}$, $10_{82}$, $10_{84}$, $10_{93}$, $10_{100}$, and $10_{152}$, as well as on the equilateral stick number of $10_{79}$. These bounds imply that the knots $10_{58}$, $10_{66}$, and $10_{80}$ have superbridge index $\le 5$, completing the project of showing that no prime knots through 10 crossings can have superbridge index larger than 5. The current best bounds on stick number and superbridge index for prime knots through 10 crossings are given in Appendix A.

We determine the action of the Kauffman bracket skein algebra of the torus on the Kauffman bracket skein module of the complement of the 3-twist knot. The point is to study the relationship between knot complements and their boundary tori, an idea that has proved very fruitful in knot theory. We place this idea in the context of Chern–Simons theory, where such actions arose in connection with the computation of the noncommutative version of the A-polynomial that was defined by Frohman, the first author, and Lofaro, but they can also be interpreted as quantum mechanical systems. Our goal is to exhibit a detailed example in a part of Chern–Simons theory where examples are scarce.

For a hyperbolic link $K$ in the thickened torus, we show there is a decomposition of the complement of a link $L$, obtained from augmenting $K$, into torihedra. We further decompose the torihedra into angled pyramids and finally angled tetrahedra. These fit into an angled structure on a triangulation of the link complement, and thus by [D. Futer and F. Guéritaud, From angled triangulations to hyperbolic structures, in Interactions between Hyperbolic Geometry, Quantum Topology and Number Theory, Contemporary Mathematics, Vol. 541 (American Mathematical Society, Providence, RI, 2011), pp. 159–182.], this shows that $L$ is hyperbolic.

We have listed all families of alternating knots with the crosscap number three.

In this paper, the authors obtain a numerical invariant $d_L$ of oriented links. It satisfies a non-homogeneous skein relation, and $d_L-1$ is additive under the connected sum operation. We give an obstruction for two links to be 6-equivalent, derived from this numerical invariant.

In this paper, we look at the knot complement problem for L-space $\mathbb{Z}$-homology spheres. We show that an L-space $\mathbb{Z}$-homology sphere $Y$ cannot be obtained as a non-trivial surgery along a knot $K\subset Y$. As a consequence, we prove that knots in an L-space $\mathbb{Z}$-homology sphere are determined by their complements.