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Abstract

Bar-Natan observed that the knots 5₁ and 10₁₃₂ have identical Jones polynomial, while recent work of Baldwin, Hu and Sivek shows that the cinquefoil 5₁ is detected by Khovanov homology. These two knots are related by a Jones-cosmetic tangle replacement, under which the (3,-2) pretzel tangle found within 10₁₃₂ is replaced by a rational tangle. The theory of immersed curves developed by Kotelskiy, Watson and Zibrowius provides us with a combinatorial means of computing reduced Bar-Natan homology, via which we investigate the existence and uniqueness of Jones-cosmetic pairs formed of a two-bridge knot and a rational tangle closure of the (3,-2) pretzel tangle with determinant less than or equal to 5. Using the observation that Jones polynomials with different spans are different, we prove that there does not exist a Jones-cosmetic pair associated with the (3,-2) pretzel tangle involving the unknot or the trefoil. Moreover, we prove that 5₁ and 10₁₃₂form the unique Jones-cosmetic pair associated with the (3,-2) pretzel tangle with determinant equal to 5.