Journal of Knot Theory and Its Ramifications

Following the classification of genus one fibered knots in lens spaces by Baker, we determine hyperbolic genus one fibered knots in lens spaces on whose all integral Dehn surgeries yield closed 3-manifolds with left-orderable fundamental groups.

We define and explore rack objects internal to categories with products. In demonstration, we classify the group-racks, and use homotopy to prove both existence and exclusion theorems for path-connected topological racks.

In this paper, we recall the geometric definition of the braid group by Emil Artin and we give a complete, elementary geometric/topological proof of the standard presentation of the braid group on $n$ strings.

We consider the relationship between the crosscap number $\gamma$ of knots and a partial order on the set of all prime knots, which is defined as follows. For two knots $K$ and $J$, we say $K\ge J$ if there exists an epimorphism $f:{\pi }_1(S^3-K)\to {\pi }_1(S^3-J)$. We prove that if $K$ and $J$ are 2-bridge knots and $K>J$, then $\gamma (K)\ge 3\gamma (J)-4$. We also classify all pairs $(K,J)$ for which the inequality is sharp. A similar result relating the genera of two knots has been proven by Suzuki and Tran. Namely, if $K$ and $J$ are 2-bridge knots and $K>J$, then $g(K)\ge 3g(J)-1$, where $g(K)$ denotes the genus of the knot $K$.

We investigate several conjectures in geometric topology by assembling computer data obtained by studying weaving knots, a doubly infinite family $W(p,n)$ of examples of hyperbolic knots. In particular, we compute some important polynomial knot invariants, as well as knot homologies, for the subclass $W(3,n)$ of this family. We use these knot invariants to conclude that all knots $W(3,n)$ are fibered knots and provide estimates for some geometric invariants of these knots. Finally, we study the asymptotics of the ranks of their Khovanov homology groups. Our investigations provide evidence for our conjecture that asymptotically as $n$ grows large, the ranks of Khovanov homology groups of $W(3,n)$ are normally distributed.

Consider a knot $K$ in $S^3$ with charge uniformly distributed on it. From the standpoint of both physics and knot theory, it is natural to try to understand the critical points of the potential and their behavior.

We show the number of critical points of the potential is at least $2t(K)+2$, where $t(K)$ is the tunnel number, defined as the smallest number of arcs one must add to $K$ such that its complement is a handlebody. The result is proven using Morse theory and stable manifold theory.

We consider a natural generalization of braids which we call shrinking braids. We state the relations of shrinking braids and use them to define algebraically the monoid $R$. We endow a subset of $R$ with a left distributive monoid structure and use it to extend the Dehornoy order on $B_{\infty }$ to an order on $R$. By using this order, we prove that $R$ is isomorphic to the monoid which is generated (geometrically) by shrinking braids.

We study Kauffman’s model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The ribbonlength is the length to width ratio of such a folded ribbon knot. We show for any knot or link type that there exist constants $c_1,c_2>0$ such that the ribbonlength is bounded above by $c_{1}Cr{(K)}^2$, and also by $c_{2}Cr{(K)}^{3/2}$. We use a different method for each bound. The constant $c_1$ is quite small in comparison to $c_2$, and the first bound is lower than the second for knots and links with $Cr(K)\le$ 12,748.