Existence of algorithms of detecting knot invertibility Are there any algorithms that detects whether the given knot is invertible or not? I don't care the algorithm is in NP or P.
 A: Given two link diagrams, there is an algorithm due to Coward and Lackenby for determining if the diagrams represent the same link. They show that given two diagrams of the same link there is a sequence of Reidemeister moves between them of size at most $\exp^{(c^n)}(n)$ (yes that's $\exp$ composed with itself $c^n$ times)  moves with $c=10^{1,000,000}$ and $n$ is the sum of the crossing numbers in the diagram. So in order to see if two link diagrams represent the same link, you just need to enumerate  all the sequences of Reidemeister moves on one of them with moves at most the bound given above. The preprint to their paper can be found here https://arxiv.org/pdf/1104.1882.pdf. 
So in order to determine if a knot is invertible you apply the above to a diagram of your knot and the diagram with the orientation reversed.
To actually show a knot is isotopic to itself with the reversed orientation is generally a challenge, and one needs to find some clever sequence of Reidemeister moves. SnapPea (https://en.wikipedia.org/wiki/SnapPea) is quite good at this.
