Knots are determined by their complements

An Introduction to Knot Theory

“…Similarly, any facet of the extensive theory of 3-dimensional manifolds can be applied to link complements; the theory of knots and links forms a fundamental source of examples in 3-manifold theory. It has recently been proved, at some length [37], that two knots with homeomorphic oriented complements are equivalent; that is not true, in general, for links of more than one component (a fairly easy exercise).…”

“…That this is not so is the substance of one of the most impressive results in knot theory of the 1980's. It is due to Gordon and J. Luecke [37] and the proof is lengthy and intricate:…”

Solving Unoriented Tangle Equations Involving 4-Plats

“…In many biological applications the knots/links involved are 4-plats, and in many of these cases, it is possible to prove that U is ambient isotopic to a sum of rational tangles and P and R are rational tangles [11,8,7,9,4,12,15]. In this situation it is possible to list all solutions for U and R as a function of P .…”

Algorithmic Topology and Classification of 3-Manifolds

“…Of course, for the case of knots one can apply the Gordon and Luecke result that knots are determined by their complements [37]. To keep the proof within reasonable limits we restrict ourselves to considering orientable 3-manifolds.…”

“…40 shows the famous Perko pair of knots, listed as distinct knots in many knot tables since the nineteenth century, until Kenneth Perko showed in 1974 that they were in fact the same knot. Less than in a minute Recognizer showed that their complements are homeomorphic (this implies that the knots are equivalent[37]). …”