Journal of Knot Theory and Its Ramifications

Finding homotopy group of spheres is an old open problem in topology. Berrick et al. derive in [A. J. Berrick, F. Cohen, Y. L. Wong and J. Wu, Configurations, braids, and homotopy groups, J. Amer. Math. Soc. 19 (2006)] an exact sequence that relates Brunnian braids to homotopy groups of spheres. We give an interpretation of this exact sequence based on the combed form for braids over the sphere developed in [R. Gillette and J. V. Buskirk, The word problem and consequences for the braid groups and mapping class groups of the two-sphere, Trans. Amer. Math. Soc. 131 (1968) 277–296] with the aim of helping one to visualize the sequence and to do calculations based on it.

Negami found an upper bound on the stick number $s(K)$ of a nontrivial knot $K$ in terms of the minimal crossing number $c(K)$: $s(K)\le 2c(K)$. Huh and Oh found an improved upper bound: $s(K)\le \frac{3}{2}(c(K)+1)$. Huh, No and Oh proved that $s(K)\le c(K)+2$ for a $2$-bridge knot or link $K$ with at least six crossings. As a sequel to this study, we present an upper bound on the stick number of Montesinos knots and links. Let $K$ be a knot or link which admits a reduced Montesinos diagram with $c(K)$ crossings. If each rational tangle in the diagram has five or more index of the related Conway notation, then $s(K)\le c(K)+3$. Furthermore, if $K$ is alternating, then we can additionally reduce the upper bound by $2$.

Let $M$ be a closed oriented 3-manifold and let $G$ be a discrete group. We consider a representation $\rho :{\pi }_1(M)\to G$. For a 3-cocycle $\alpha$, the Dijkgraaf–Witten invariant is given by $({\rho }^{\ast }\alpha )[M]$, where ${\rho }^{\ast }:H^3(G)\to H^3(M)$ is the map induced by $\rho$, and $[M]$ denotes the fundamental class of $M$. Note that $({\rho }^{\ast }\alpha )[M]=\alpha ({\rho }_{\ast }[M])$, where ${\rho }_{\ast }:H_3(M)\to H_3(G)$ is the map induced by $\rho$, we consider an equivalent invariant ${\rho }_{\ast }[M]\in H_3(G)$, and we also regard it as the Dijkgraaf–Witten invariant. In 2004, Neumann described the complex hyperbolic volume of $M$ in terms of the image of the Dijkgraaf–Witten invariant for $G={\mathrm{\text{SL}}}_2ℂ$ by the Bloch–Wigner map from $H_3({\mathrm{\text{SL}}}_2ℂ)$ to the Bloch group of $ℂ$.

In this paper, by replacing $ℂ$ with a finite field ${𝔽}_p$, we calculate the reduced Dijkgraaf–Witten invariants of the complements of twist knots, where the reduced Dijkgraaf–Witten invariant is the image of the Dijkgraaf–Witten invariant for SL${}_2{𝔽}_p$ by the Bloch–Wigner map from $H_3({\mathrm{\text{SL}}}_2{𝔽}_p)$ to the Bloch group of ${𝔽}_p$.

In this paper, we study Dehn colorings of spatial graph diagrams, and classify the vertex conditions, equivalently the palettes. We give some example of spatial graphs which can be distinguished by the number of Dehn colorings with selecting an appropriate palette. Furthermore, we also discuss the generalized version of palettes, which is defined for knot-theoretic ternary-quasigroups and region colorings of spatial graph diagrams.

A twisted torus knot $T(p,q,r,s)$ is a torus knot $T(p,q)$ with $r$ adjacent strands twisted fully $s$ times. In this paper, we determine the braid index of the knot $T(p,q,r,s)$ when the parameters $p,q,r$ satisfy $1<q<p<r\le p+q$. If the last parameter $s$ additionally satisfies $s=-1$, then we also determine the parameters $p,q,r$ for which $T(p,q,r,s)$ is a torus knot.

We introduce a numerical invariant called the braid alternation number that measures how far a link is from being an alternating closed braid. This invariant resembles the alternation number, which was previously introduced by the second author. However, these invariants are not equal, even for alternating links.

We study the relation of this invariant with others and calculate this invariant for some infinite knot families. In particular, we show arbitrarily large gaps between the braid alternation number and the alternation and unknotting numbers. Furthermore, we estimate the braid alternation number for prime knots with nine crossings or less.

Extending upon our previous work, we verify the Jones Unknot Conjecture for all knots up to 24 crossings. We describe the method of our approach and analyze the growth of the computational complexity of its different components.

Using a combinatorial argument, we prove the well-known result that the Wirtinger and Dehn presentations of a link in 3-space describe isomorphic groups. The result is not true for links $\ell$ in a thickened surface $S\times [0,1]$. Their precise relationship, as given in [R. E. Byrd, On the geometry of virtual knots, M.S. Thesis, Boise State University (2012)], is established here by an elementary argument. When a diagram in $S$ for $\ell$ can be checkerboard shaded, the Dehn presentation leads naturally to an abelian “Dehn coloring group,” an isotopy invariant of $\ell$. Introducing homological information from $S$ produces a stronger invariant, ${𝒞}$, a module over the group ring of $H_1(S;\mathbb{Z})$. The authors previously defined the Laplacian modules ${{ℒ}}_G,{{ℒ}}_{G^{\ast }}$ and polynomials ${\mathrm{\Delta }}_G,{\mathrm{\Delta }}_{G^{\ast }}$ associated to a Tait graph $G$ and its dual $G^{\ast }$, and showed that the pairs $\{{{ℒ}}_G,{{ℒ}}_{G^{\ast }}\}$, $\{{\mathrm{\Delta }}_G,{\mathrm{\Delta }}_{G^{\ast }}\}$ are isotopy invariants of $\ell$. The relationship between ${𝒞}$ and the Laplacian modules is described and used to prove that ${\mathrm{\Delta }}_G$ and ${\mathrm{\Delta }}_{G^{\ast }}$ are equal when $S$ is a torus.

In this paper, we show that every spinal open book decomposition of a closed oriented $3$-manifold $M$ spinal open book embeds into a certain spinal open book decomposition of ${\#}_k{S}^2\tilde{\times }{S}^3$, the connected sum of $k$ copies of the twisted $S^3$-bundle over $S^2$, where $k$ depends on the spinal open book decomposition of $M$. We also discuss spinal open book embeddings of a huge class of spinal open books of closed oriented $3$-manifolds into the trivial spinal open book of the $5$-sphere $S^5$. Finally, we show that given a closed oriented $3$-manifold $M$, there exists a spinal open book for $M$ such that $M$ spinal open book embeds into the trivial spinal open book of $S^5.$ In particular, this gives another proof of Hirsch’s theorem which states that every closed orientable $3$-manifold embeds in $S^5.$