Abstract
Let K be a knot in S3 that admits two disjoint, separating, essential surfaces with boundary, where one of them has an accidental annulus. Then, K admits infinitely many closed essential surfaces unless the annulus is "centered". This implies for example that any non-satellite knot that admits a 4-puntured essential sphere and any alternating knot that admits a closed essential surface of genus 2 admit infinitely many closed essential surfaces.
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