Related papers: On surgery curves for genus one slice knots

We give an example of a 3-component smoothly slice boundary link, each of whose components has a genus one Seifert surface, such that any metaboliser of the boundary link Seifert form is represented by 3 curves on the Seifert surfaces that…

The concordance genus of a knot K is the minimum three-genus among all knots concordant to K. For prime knots of 10 or fewer crossings there have been three knots for which the concordance genus was unknown. Those three cases are now…

We construct many examples of non-slice knots in 3-space that cannot be distinguished from slice knots by previously known invariants. Using Whitney towers in place of embedded disks, we define a geometric filtration of the 3-dimensional…

Using an obstruction based on Donaldson's theorem, we derive strong restrictions on when a Seifert fibered space $Y = F(e; \frac{p_1}{q_1}, \ldots, \frac{p_k}{q_k})$ over an orientable base surface $F$ can smoothly embed in $S^4$. This…

We show that the Vassiliev invariants of a knot K, are obstructions to finding a regular Seifert surface, S, whose complement looks "simple" (e.g. like the complement of a disc) to the lower central series of its fundamental group.

Recent work of Jonathan Campbell and Inna Zakharevich has focused on building machinery for studying scissors congruence problems via algebraic $K$-theory, and applying these tools to studying the Grothendieck ring of varieties. In this…

We construct an infinite family of topologically slice knots that are not smoothly concordant to their reverses. More precisely, if T denotes the concordance group of topologically slice knots and R is the involution of T induced by string…

We describe a method to compute the Culler-Shalen seminorms of a knot, using the (-3,3,4) pretzel knot as an illustrative example. We deduce that the SL2(C)-character variety of this knot consists of three algebraic curves and that it…

We use the Blanchfield form to obtain a lower bound on the equivariant slice genus of a strongly invertible knot. For our main application, let $K$ be a genus one strongly invertible slice knot with nontrivial Alexander polynomial. We show…

We study cosmetic crossings in knots of genus one and obtain obstructions to such crossings in terms of knot invariants determined by Seifert matrices. In particular, we prove that for genus one knots the Alexander polynomial and the…

Let $M_K$ be the 2-fold branched cover of a knot $K in $S^3$. If $H_1(M_K) = {\bf Z}_3 \oplus {\bf Z}_{3^{2i}} \oplus G$ where 3 does not divide the order of $G$ then $K$ is not of order 4 in the concordance group. This obstruction detects…

We prove a recent conjecture of Manolescu-Willis which states that the $s$-invariant of a knot in $\mathbb{RP}^3$ (as defined by them) gives a lower bound on its null-homologous slice genus in the unit disk bundle of $TS^2$. We also…

Suppose C is a singular curve in CP^2 and it is topologically an embedded surface of genus g; such curves are called cuspidal. The singularities of C are cones on knots K_i. We apply Heegaard Floer theory to find new constraints on the sets…

We study the classification of slice disks of knots up to isotopy and diffeomorphism using an invariant in knot Floer homology. We compute the invariant of a slice disk obtained by deform-spinning, and show that it can be effectively used…

A Seifert surface for a knot K is called canonical if it can be built by applying Seifert's algorithm to some projection of K. The canonical genus of K is the smallest genus of a surface so obtained. In this paper we show that there is a…

Let $K$ be a knot in the 3-sphere, viewed as the ideal boundary of hyperbolic 4-space $\mathbb{H}^4$. We prove that the number of minimal discs in $\mathbb{H}^4$ with ideal boundary $K$ is a knot invariant. I.e.\ the number is finite and…

Let $(S,H)$ be a general primitively polarized $K3$ surface of genus $\p$ and let $p_a(nH)$ be the arithmetic genus of $nH.$ We prove the existence in $|\mathcal O_S(nH)|$ of curves with a triple point and $A_k$-singularities. In…

Let $K$ be a knot in an L-space $Y$ with a Dehn surgery to a surface bundle over $S^1$. We prove that $K$ is rationally fibered, that is, the knot complement admits a fibration over $S^1$. As part of the proof, we show that if $K\subset Y$…

We present a framework for studying transverse knots and symplectic surfaces utilizing the Seiberg-Witten monopole equation. Our primary approach involves investigating an equivariant Seiberg-Witten theory introduced by Baraglia-Hekmati on…

We construct an invariant of virtual knots which is a sliceness obstruction and sensitive to the $\Delta$-move. This invariants works if $\Z_{2}\oplus \Z_{2}$-index of chords is present.