# obtaining circle bundle over torus by trefoil surgery

Does any integer surgery on a right or left trefoil knot give the $S^1$-bundle over $T^2$ with Euler number $1$?

No. The result of any surgery on a knot has cyclic first homology, whereas this torus bundle has first homology of rank 2.

Jonathan's answer above is direct and complete. This answer is meant to give a less elegant proof but with more details on trefoil surgeries.

Following Louise Moser's classification of surgeries on torus knot, Proposition 3.1 of:

Moser, Louise. "Elementary surgery along a torus knot." Pacific Journal of
Mathematics 38.3 (1971): 737-745.

there is a 1-parameter family of $S^1$ bundles over $T^2$ that can be obtained from the right handed trefoil knot (same for left). In this case, the surgery parameters should be $p=6-6q$ (q relatively prime to 6. Also, Moser's convention for p and q are opposite the more popular convention that a meridian is 1/0). Also, by Proposition 3.1 mentioned above, these are the only surgeries on the trefoil knot that are torus bundles. However, all of these bundles have cyclic homology so they can't be of the form asked for by the OP.
 Kadokami, Teruhisa, and Masafumi Shimozawa. "Dehn surgery along torus links."