Journal of Knot Theory and Its Ramifications

We enumerate ribbon 2-knots presented by virtual arc diagrams with up to four classical crossings. We use a linear Gauss diagram for a virtual arc diagram.

An axis of a link projection is a closed curve which lies symmetrically on each region of the link projection. In this paper we define axis systems of link projections and characterize axis systems of the standard projections of twist knots.

Let $t_{\alpha ,\beta }\subset S^2\times S^1$ be an ordinary fiber of a Seifert fibering of $S^2\times S^1$ with two exceptional fibers of order $\alpha$. We show that any Seifert manifold with Euler number zero is a branched covering of $S^2\times S^1$ with branching $t_{\alpha ,\beta }$ if $\alpha \ge 3$. We compute the Seifert invariants of the Abelian covers of $S^2\times S^1$ branched along a $t_{\alpha ,\beta }$. We also show that $t_{2,1}$, a non-trivial torus knot in $S^2\times S^1$, is not universal.

We discuss the connection between colorings of a link diagram and the Goeritz matrix.

The structure of classical minimal prime knot presentations suggests that there are often, perhaps always, subsegments that present either the trefoil or the figure-eight knot. A comprehensive study of the subknots of the minimal prime knot presentations through 15 crossings shows that this is always the case for these knot presentations. Among this set of 313, 258 prime knot presentation, there are only 547, or 0.17%, that do not contain a trefoil subknot. Thus, 99.83% of minimal prime knot presentations through 15 crossings contain trefoil subknots. We identify several infinite minimal alternating prime knot families that do not contain trefoil subknots but always contain figure-eight knots. We discuss the statistics of subknots of prime knots and, using knot presentation fingerprints, illustrate the complex character of the subknots of these classic minimal prime knot presentations. We conclude with a discussion the conjectures and open questions that have grown out of our research.

We prove that for any odd $n\ge 3$, the $n$-palette number of any effectively $n$-colorable $2$-bridge knot is equal to $2+\lfloor {log}_{2}n\rfloor$. Namely, there is an effectively $n$-colored diagram of the $2$-bridge knot such that the number of distinct colors that appeared in the diagram is exactly equal to $2+\lfloor {log}_{2}n\rfloor$.

We introduce dual graph diagrams representing oriented knots and links. We use these combinatorial structures to define corresponding algebraic structures which we call biquasiles whose axioms are motivated by dual graph Reidemeister moves, generalizing the Dehn presentation of the knot group analogously to the way quandles and biquandles generalize the Wirtinger presentation. We use these structures to define invariants of oriented knots and links and provide examples.

We prove that the Torelli group of an oriented surface with any number of boundary components is at least exponentially distorted in the mapping class group by using Broaddus–Farb–Putman’s techniques. Further we show that the distortion of the Torelli group in the level $d$ mapping class group is the same as that in the mapping class group.

We investigate properties of spatial graphs on the standard torus. It is known that nontrivial embeddings of planar graphs in the torus contain a nontrivial knot or a nonsplit link due to [2, 3]. Building on this and using the chirality of torus knots and links [9, 10], we prove that the nontrivial embeddings of simple 3-connected planar graphs in the standard torus are chiral. For the case that the spatial graph contains a nontrivial knot, the statement was shown by Castle et al. [5]. We give an alternative proof using minors instead of the Euler characteristic. To prove the case in which the graph embedding contains a nonsplit link, we show the chirality of Hopf ladders with at least three rungs, thus generalizing a theorem of Simon [12].

Some of the results on welded knots in the title paper were incorrect. In this corrigendum, these are corrected and extended. An additional appendix is also included.