Journal of Knot Theory and Its Ramifications

For a knot, the ascending number is the minimum number of crossing changes which are needed to obtain an ascending diagram. We study the ascending number of a knot by analyzing the Conway polynomial. In this paper, we give a sharper lower bound of the ascending number of a knot and newly determine the ascending number for 26 prime knots up to 10 crossings.

In the present paper, we consider two types of 2-component links with genus two Heegaard splittings. One of them is an ordinary tunnel number one link, and the other is a somewhat different tunnel number one link. We will try to detect the differences between those two types. In fact, we will characterize composite tunnel number one links of the second type, and tunnel number one links of the second type with essential tori.

We propose a method to determine the character variety of a class $J(m,n)$ of rational knots, which includes the twist knots. The defining polynomials depend only on the variables $m$ and $n$. This answers for these classes of knots a question posed in [H. M. Hilden, M. T. Lozano and J. M. Montesinos–Amilibia, On the character variety of group representations of a 2-bridge link $p/3$ into $PSL(2,ℂ)$, Bol. Soc. Mat. Mexicana 37(2) (1992) 241–262], and allows us to give an easy geometrical description of the considered character variety. Our results are obtained by using special presentations of the knot groups whose relators are palindromes.

The Thurston norm of a three-manifold measures the complexity of surfaces representing two-dimensional homology classes. We study the possible unit balls of Thurston norms of three-manifolds $M$ with $b_1(M)=2$, and whose fundamental groups admit presentations with two generators and one relator. We show that even among this special class, there are three-manifolds such that the unit ball of the Thurston norm has arbitrarily many faces.

In this paper, we combine the recent four-dimensional light bulb theorem of David Gabai and a construction of concordances for knots in $S^2\times S^1$ due to Eylem Zeliha Yildiz to construct an explicit concordance between the standard surface of genus $g$ in $S^2\times S^2$ and any homologous surface using elementary methods.

A checkerboard graph of a special diagram of an oriented link is made a directed, edge-weighted graph in a natural way so that a principal submatrix of its Laplacian matrix is a Seifert matrix of the link. Doubling and weighting the edges of the graph produces a second Laplacian matrix such that a principal submatrix is an Alexander matrix of the link. The Goeritz matrix and signature invariants are obtained in a similar way. A device introduced by Kauffman makes it possible to apply the method to general diagrams.

For a knot $K\subset S^3$, its exterior $E(K)=S^3\setminus \eta (K)$ has a singular foliation by Seifert surfaces of $K$ derived from a circle-valued Morse function $f:E(K)\to S^1$. When $f$ is self-indexing and has no critical points of index 0 or 3, the regular levels that separate the index-1 and index-2 critical points decompose $E(K)$ into a pair of compression bodies. We call such a decomposition a circular Heegaard splitting of $E(K)$. We define the notion of circular distance (similar to Hempel distance) for this class of Heegaard splitting and show that it can be bounded under certain circumstances. Specifically, if the circular distance of a circular Heegaard splitting is too large: (1) $E(K)$ cannot contain low-genus incompressible surfaces, and (2) a minimal-genus Seifert surface for $K$ is unique up to isotopy.

A partial order on prime knots can be defined by declaring $J\ge K$, if there exists an epimorphism from the knot group of $J$ onto the knot group of $K$. Suppose that $J$ is a 2-bridge knot that is strictly greater than $m$ distinct, nontrivial knots. In this paper, we determine a lower bound on the crossing number of $J$ in terms of $m$. Using this bound, we answer a question of Suzuki regarding the 2-bridge epimorphism number $\text{EK}(n)$ which is the maximum number of nontrivial knots which are strictly smaller than some 2-bridge knot with crossing number $n$. We establish our results using techniques associated with parsings of a continued fraction expansion of the defining fraction of a 2-bridge knot.