How can I verify a solution to Travelling Salesman Problem in polynomial time?
The crux is that you have to consider the decision problem:


  Travelling Salesman Problem (Decision Version). Given a weighted graph G and a target cost C, is there a Hamiltonian cycle in G whose weight is at most C?


For a 'yes' instance, the certificate is just some Hamiltonian cycle whose weight is at most C. If you could solve this problem efficiently, you could find the cost of a minimum tour by binary search, starting with the weight of the entire network as an upper bound.