Journal of Knot Theory and Its Ramifications

A ribbon is a smooth mapping (possibly self-intersecting) of an annulus $S^1\times I$ in 3-space having constant width $R$. Given a regular parametrization $x(s)$, and a smooth unit vector field $u(s)$ based along $x$, for a knot $K$, we may define a ribbon of width $R$ associated to $x$ and $u$ as the set of all points $x(s)+ru(s)$, $r\in [0,R]$. For large $R$, ribbons, and their outer edge curves, may have self-intersections. In this paper, we analyze how the knot type of the outer ribbon edge $x(s)+Ru(s)$ relates to that of the original knot $K$. Generically, as $R\to \infty$, there is an eventual constant knot type. This eventual knot type is one of only finitely many possibilities which depend just on the vector field $u$. The particular knot type within the finite set depends on the parametrized curves $x(s)$, $u(s)$, and their interactions. We demonstrate a way to control the curves and their parametrizations so that given two knot types $K_1$ and $K_2$, we can find a smooth ribbon of constant width connecting curves of these two knot types.

In 1947, in the paper “Theory of Braids,” Artin raised the question of whether isotopy and homotopy of braids on the disk coincide. Twenty seven years later, Goldsmith answered his question: she proved that in fact the group structures are different, exhibiting a group presentation and showing that the homotopy braid group on the disk is a proper quotient of the Artin braid group on the disk $B_n$, denoted by ${\widehat{B}}_n$. In this paper, we extend Goldsmith’s answer to Artin for closed, connected and orientable surfaces different from the sphere. More specifically, we define the notion of homotopy generalized string links on surfaces, which form a group which is a proper quotient of the braid group on a surface $B_n(M)$, denoting it by ${\widehat{B}}_n(M)$. We then give a presentation of the group ${\widehat{B}}_n(M)$ and find that the Goldsmith presentation is a particular case of our main result, when we consider the surface $M$ to be the disk. We close with a brief discussion surrounding the importance of having such a fixed construction available in the literature.

A knot in a thickened surface $K$ is a smooth embedding $K:S^1\to \mathrm{\Sigma }\times [0,1]$, where $\mathrm{\Sigma }$ is a closed, connected, orientable surface. There is a bijective correspondence between knots in $S^2\times [0,1]$ and knots in $S^3$, so one can view the study of knots in thickened surfaces as an extension of classical knot theory. An immediate question is if other classical definitions, concepts, and results extend or generalize to the study of knots in a thickened surface. One such famous result is the Fox–Milnor Theorem, which relates the Alexander polynomials of concordant knots. We prove a Fox–Milnor Theorem for concordant knots in a thickened surface by using Milnor torsion.

Satoh and Taniguchi introduced the $n$-writhe $J_n$ for each non-zero integer $n$, which is an invariant for virtual knots. The $n$-writhes give the coefficients of some polynomial invariants for virtual knots including the index polynomial, the odd writhe polynomial and the affine index polynomial. It is obvious that the virtualization of a real crossing is an unknotting operation for virtual knots. The values of $n$-writhes changed by some local moves are calculated. However for the virtualization, it is unknown. In this paper, we show that for any given non-zero integer $n$ and any given integer $N$, there exists a virtual knot whose unknotting number by the virtualization is one and the value of the $n$-writhe equals $N$. Namely, the virtualization of a real crossing changes the value of $n$-writhe by any given integer $N$.

We introduce multi-tribrackets, algebraic structures for region coloring of diagrams of knots and links with different operations at different kinds of crossings. In particular, we consider the case of component multi-tribrackets which have different tribracket operations at single-component crossings and multi-component crossings. We provide examples to show that the resulting counting invariants can distinguish links which are not distinguished by the counting invariants associated to the standard tribracket coloring. We reinterpret the results of [S. Nelson and S. Pico. Virtual tribrackets, preprint (2018), arXiv:1803.03210] in terms of multi-tribrackets and consider future directions for multi-tribracket theory.

A directed graph $G$ is intrinsically linked if every embedding of that graph contains a nonsplit link $L$, where each component of $L$ is a consistently oriented cycle in $G$. A tournament is a directed graph where each pair of vertices is connected by exactly one directed edge.

We consider intrinsic linking and knotting in tournaments, and study the minimum number of vertices required for a tournament to have various intrinsic linking or knotting properties. We produce the following bounds: intrinsically linked ($n=8$), intrinsically knotted ($9\le n\le 12$), intrinsically 3-linked ($10\le n\le 23$), intrinsically 4-linked ($12\le n\le 66$), intrinsically 5-linked ($15\le n\le 154$), intrinsically $m$-linked ($3m\le n\le 8{(2m-3)}^2$), intrinsically linked with knotted components ($9\le n\le 107$), and the disjoint linking property ($12\le n\le 14$).

We also introduce the consistency gap, which measures the difference in the order of a graph required for intrinsic $n$-linking in tournaments versus undirected graphs. We conjecture the consistency gap to be nondecreasing in $n$, and provide an upper bound at each $n$.

Margalit and Schleimer observed that Dehn twists on orientable surfaces have nontrivial roots. We investigate the problem of roots of a Dehn twist $t_c$ about a nonseparating circle $c$ in the mapping class group ${ℳ}(N_g)$ of a nonorientable surface $N_g$ of genus $g$. We explore the existence of roots and, following the work of McCullough, Rajeevsarathy and Monden, give a simple arithmetic description of their conjugacy classes. We also study roots of maximal degree and prove that if we fix an odd integer $n>1$, then for each sufficiently large $g$, $t_c$ has a root of degree $n$ in ${ℳ}(N_g)$. Moreover, for any possible degree $n$, we provide explicit expressions for a particular type of roots of Dehn twists about nonseparating circles in $N_g$.

In [D. Auckly, H. J. Kim, P. Melvin and D. Ruberman, Equivariant corks, Algebr. Geom. Topol. 17(3) (2017) 1771–1783], Auckly–Kim–Melvin–Ruberman showed that for any finite subgroup $G$ of $SO(4)$ there exists a contractible smooth 4-manifold with an effective $G$-action on its boundary so that the twists associated to the non-trivial elements of $G$ don’t extend to diffeomorphisms of the entire manifold. We give a different proof of this phenomenon using the Heegaard Floer techniques in [S. Akbulut and Ç. Karakurt, Action of the cork twist on Floer homology, in Proc. Gökova Geometry-Topology Conf. 2011 (International Press, Somerville, MA, 2012), pp. 42–52].

We introduce the notion of a “state function” for framed tangles in a disk. After choosing a finite set of states for each marked disk, a state function is a projection from the vector space spanned by all tangles to the vector space spanned by the states, that is local, and topologically invariant. Given the states for the Kauffman bracket, and the quantum $SU(3)$-invariant we classify all state functions, and then compare our results to the literature.

We extend the theory of hyperbolicity of links in the 3-sphere to tg-hyperbolicity of virtual links, using the fact that the theory of virtual links can be translated into the theory of links living in closed orientable thickened surfaces. When the boundary surfaces are taken to be totally geodesic, we obtain a tg-hyperbolic structure with a unique associated volume. We prove that all virtual alternating links are tg-hyperbolic. We further extend tg-hyperbolicity to several classes of non-alternating virtual links. We then consider bounds on volumes of virtual links and include a table for volumes of the 116 nontrivial virtual knots of four or fewer crossings, all of which, with the exception of the trefoil knot, turn out to be tg-hyperbolic.