I'm referring to the decision version of both $TSP$ and $HamCycle$. The first is, given a graph $G=(V,E)$, a weight function $w:E\rightarrow \mathbb R^+$ and an integer $k$, is there a simple cycle with weight $\leq k$ which passes, once, through every vertex? (The weight of the cycle is the sum of the weights of the edges). The second problem is finding a Hamiltoninan Cycle (a cycle that passes through every vertex only once).
I know both are $NPC$. I am interested in finding reduction between both these problems. The first reduction $HamCycle \leq TSP$ is quite easy: for every edge in the original graph you give a weight $0$ and for every pair of vertices who did not have an edge you give a weight of $1$. Then you ask whether the agent can travel with sum of path $\leq 0$. The correctness of this follows pretty quickly.
However, my question if for the reduction the other way around: $TSP\leq HamCycle$. I could not come up with an idea for such. I am aware that since both are $NPC$ there exists a reduction through Cook-Levin Theorem and $SAT$, but I am wondering if there is a direct reduction $TSP\leq HamCycle$?