Related papers: Average four-genus of two-bridge knots

We prove that the signature bound for the topological 4-genus of 3-strand torus knots is sharp, using McCoy's twisting method. We also show that the bound is off by at most 1 for 4-strand and 6-strand torus knots, and improve the upper…

We prove that for 2-bridge knots, the diameter, D, of the set of boundary slopes is twice the crossing number, c. This constitutes partial verification of a conjecture that, for all knots in S^3, D is at most 2c.

We classify 3-braid knots whose topological 4-genus coincides with their Seifert genus, using McCoy's twisting method and the Xu normal form. In addition, we give upper bounds for the topological 4-genus of positive and strongly…

Let $K,K'$ be two-bridge knots of genus $n,k$ respectively. We show the necessary and sufficient condition of $n$ in terms of $k$ that there exists an epimorphism from the knot group of $K$ onto that of $K'$.

We adapt Seifert's algorithm for classical knots and links to the setting of tri-plane diagrams for bridge trisected surfaces in the 4-sphere. Our approach allows for the construction of a Seifert solid that is described by a Heegaard…

We study the equivariant concordance classes of two-bridge knots, providing an easy formula to compute their butterfly polynomial, and we give two different proofs that no two-bridge knot is equivariantly slice. Finally, we introduce a new…

For a given knot, we study the minimal number of positive eigenvalues of the double branched cover over spanning surfaces for the knot. The value gives a lower bound for various genera, the dealternating number and the alternation number of…

We classify all knot diagrams of genus two and three, and give applications to positive, alternating and homogeneous knots, including a classification of achiral genus 2 alternating knots, slice or achiral 2-almost positive knots, a proof…

Let $K$ be a knot with an unknotting tunnel $\gamma$ and suppose that $K$ is not a 2-bridge knot. There is an invariant $\rho = p/q \in \mathbb{Q}/2 \mathbb{Z}$, $p$ odd, defined for the pair $(K, \gamma)$. The invariant $\rho$ has…

We give examples of knots with some unusual properties of the crossing number of positive diagrams or strand number of positive braid representations. In particular we show that positive braid knots may not have positive minimal (strand…

We give infinitely many examples of 2-bridge knots for which the topological and smooth slice genera differ. The smallest of these is the 12-crossing knot $12a255$. These also provide the first known examples of alternating knots for which…

For each three-bridge link of a certain form, we construct a taut Seifert surface for the link and establish whether the link is fibred. Using this, we also give the genus and fibredness of satellite knots whose pattern is constructed from…

This paper, to be regularly updated, lists those prime knots with the fewest possible number of crossings for which values of basic knot invariants, such as the unknotting number or the smooth 4-genus, are unknown. This list is being…

We investigate the nonorientable 4-genus $\gamma_4$ of a special family of 2-bridge knots, the twist knots and double twist knots $C(m,n)$. Because the nonorientable 4-genus is bounded by the nonorientable 3-genus, it is known that…

We show that a positive braid knot has maximal topological 4-genus exactly if it has maximal signature invariant. As an application, we determine all positive braid knots with maximal topological 4-genus and compute the topological 4-genus…

This paper contains the results of efforts to determine values of the smooth and the topological slice genus of 11- and 12-crossing knots. Upper bounds for these genera were produced by using a computer to search for genus one concordances…

For every integer g, we construct a 2-solvable and 2-bipolar knot whose topological 4-genus is greater than g. Note that 2-solvable knots are in particular algebraically slice and have vanishing Casson-Gordon obstructions. Similarly all…

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 $K \geq J$ if there exists an epimorphism…

In this note, we prove a lower bound for the positive kinkiness of a closed braid which we then use to derive an estimate for the positive kinkiness of a link in terms of its Seifert system. As an application, we show that certain pretzel…

A marked strongly invertible knot is a triple $(K,h,\delta)$ of a knot $K$ in $S^3$, a strong inversion $h$ of $K$, and a subarc $\delta \subset \operatorname{Fix}(h)\cong S^1$ bounded by $\operatorname{Fix}(h)\cap K\cong S^0$. An invariant…