Can we construct a Karp reduction from a Cook reduction between NP problems?
its a generally open TCS problem subject to ongoing research whether &amp; the exact conditions Cook &amp; Karp reductions are equivalent and is apparently closely related to the open NP=?coNP question &amp; other complexity class separations eg E=?NE (wrt sparse languages).

here are two research papers on the subject &amp; further leads on tcs.se via similar question:


Are Cook and Karp Ever the Same? Beigel, Fortnow
Cook vs. Karp-Levin: Seperating Completeness Notions If NP Is Not Small (1995) Lutz, Mayordomo
many one reductions vs Turing reductions to define NPC, tcs.se