Each crossing in a link diagram of a link $L$ has an A-resolution and a B-resolution.
Resolving all crossings gives a collection of circles in the plane, connected by dotted lines. A state of a link diagram is a choice for each crossing of either an A-resolution or a B-resolution. The resulting state graph is the graph whose vertices correspond to the circles, and whose edges correspond to dotted lines. A link diagram is said to be semi-adequate if either the state-graph resulting from A-resolving all crossings, or the state-graph resulting from B-resolving all crossings (or both) have no single-edge loops.
In Section 1.3 of Guts of surfaces and the colored Jones polynomial, the authors state that, "The condition that a link be semi-adequate seems to be rather mild". As evidence, they point out several low-crossing-number computations, including a computation of Stoimenow that shows that at least 249,649 of the 253,293 prime knots with 15 crossings are semi-adequate.
The question which lept to my mind (and I have no idea whether this is trivial, well-known, hard, or open) is whether this is an instance of the Strong Law of Small Numbers (which perhaps in this context should be called the `strong law of small links'):
Question: Is a generic link diagram semi-adequate?
When I say generic, I mean in a Kolmogorov complexity sense. Perhaps we choose a random 4-valent (planar) graph, and flip a coin to decide whether each crossing should be over or under.