Ambiguity in the unoriented knot connected sum It is well known that there might be ambiguity in the unoriented knot connected sum if the knots concerned are not invertible.
E.g., consider 8_17, the only knot with crossing number 8 which is non-invertible.
On page 12 of Adams' knot book, he claimed that there arise two inequivalent knots when we do connected sum of 8_17 with itself.

No detail is given. How do knot theorists distinguish the two knots?
 A: This was done by Schubert, in 1949 "Die eindeutige Zerlegbarkeit eines Knotens in Primknoten". In his original proof he uses a decomposition of pairs $(^3,)$ using embedded spheres with two marked points. In his later paper "Knoten und vollringe" (1953) he noticed a more efficient proof using knot exteriors and what we now call incompressible tori. This paper marks when people realized that many problems in knot theory should probably be approached by studying knot exteriors and using 3-manifold theory techniques, i.e. the merger of knot theory and 3-manifold theory begins here.
Schubert's theorem states that oriented knots in $^3$, taken up to isotopy, when using the connect-sum operation, this is a free commutative monoid. So the ambiguity Adams refers to is controlled by Schubert's theorem. Other than Schubert's theorem you need to know which prime knots are invertible. There are many ways to answer this latter question.  Here are a few.

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*Geometrization provides an answer to the question of which knots are invertible.  You can see this from the Epstein-Penner decomposition of a hyperbolic knot exterior.  Software like Snappea computes this, so it is a quick and complete check for hyperbolic knots.


*If the knot isn't hyperbolic, geometrization gives a slightly more involved answer.  There is a writeup in my "JSJ-decompositions of knot and link complements in $S^3$" paper.


*You can rule out invertibility using Waldhausen-type techniques, considering automorphisms of the fundamental group of the knot exterior that invert a meridian.  i.e. this is a purely group-theoretic approach.  I believe this was Hartley's approach in 1983.
Those are some techniques that come to mind.
I believe $8_{17}$ was the first-discovered non-invertible knot.
