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

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

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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.
http://projecteuclid.org/download/pdf_1/euclid.pjm/1102969920 

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.

In order to find such examples, one could use surgeries on torus links. A great reference in this case is:

 Kadokami, Teruhisa, and Masafumi Shimozawa. "Dehn surgery along torus links."
 Journal of Knot Theory and Its Ramifications 19.04 (2010): 489-502.
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Finally it is worth pointing out that the desired manifold can arise as surgery on links which are not torus links as well.

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