Almost positive links are strongly quasipositive

Abstract

We prove that any link admitting a diagram with a single negative crossing is strongly quasipositive. This answers a question of Stoimenow's in the (strong) positive. As a second main result, we give a simple and complete characterization of link diagrams with quasipositive canonical surface (the surface produced by Seifert's algorithm). As applications, we determine which prime knots up to 13 crossings are strongly quasipositive, and we confirm the following conjecture for knots that have a canonical surface realizing their genus: a knot is strongly quasipositive if and only if the Bennequin inequality is an equality.

Keywords: 57M25.

Fig. 1

Inserting a positive crossing next to another one by positive Hopf plumbing. Red and blue indicate the two sides of oriented surfaces. Dotted lines are hidden below a surface. a Two Seifert circles connected by a positive crossing. The small arrows indicate positive normal vectors of the surfaces. b Surface obtained from a by plumbing a positive Hopf band along the gray curve on the positive side of the surface. c This surface is isotopic to b (pull the Hopf band away from the crossing). Note that the central white region could contain infinity

Fig. 2

Top and left: a type I diagram D (the $(-3,-3,1)$-pretzel diagram) and $\mathrm{\Sigma }(D)$. Below: its Seifert graph, with the unique edge of weight $-1$ drawn dashed. On the right: a diagram $D^{\prime }$ obtained from D by applying (7) to the closed interval drawn gray and dotted, the surface $\mathrm{\Sigma }(D^{\prime })$, and the graph $\mathrm{\Gamma }(D^{\prime })$

Fig. 3

How to find an interval as in (7) (drawn green)

Fig. 4

$s_1$ and $s_2$ cut the plane into four regions

Fig. 5

Left: crossings c and $c^{\prime }$ next to each other on k. Middle and right: crossings c and $c^{\prime }$ next to each other

Fig. 6

Swapping the crossing c, which is adjacent to $s_1$ and $s_2$. Note that no other Seifert circles or crossings are present in the disk where the modification occurs

Fig. 7

Diagrams to which (2’)–(6’) do not apply

Fig. 8

k is $s_1$ or $s_2$ and c is next to one of the two intersection points of $s_1$ and $s_2$ (bottom)

Fig. 9

Left-to-middle and right-to-middle: swapping a Seifert circle over $c_{+}$ into a crossing. Left-to-right: swapping a Seifert circle from $O_1$ over $c_{+}$ into a Seifert circle in $O_2$

Fig. 10

Inserting a positive crossing next to another one by positive Hopf plumbing (generalizing Fig. 1). a Local picture of ${\mathrm{\Sigma }}^{\prime }(D)$ containing the two ribbons corresponding to two crossings $c_1$ and $c_2$ that are next to each other. Note that the central white region could contain infinity. b The result of plumbing a positive Hopf band in c. c A closed interval (gray) in ${\mathrm{\Sigma }}^{\prime }(D^{\prime })$ along which a positive Hopf band gets plumbed to the blue side

Fig. 11

A diagram of the situation of Lemma 3.15