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Comparing the behaviors of the Seifert genus $g(K_n)$ and the slice genus $g_4(K_n)$ under twistings, we prove that if $g(K_n) - g_4(K_n) < C$ for some constant $C$ for infinitely many integers $n > 0$ or $g(K_n) / g_4(K_n) \\to 1$ as $n \\to \\infty$, then either the winding number of $K$ about $c$ is zero or the winding number equals the wrapping number. 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Baker, Kimihiko Motegi","submitted_at":"2017-05-29T19:40:50Z","abstract_excerpt":"Twisting a knot $K$ in $S^3$ along a disjoint unknot $c$ produces a twist family of knots $\\{K_n\\}$ indexed by the integers. Comparing the behaviors of the Seifert genus $g(K_n)$ and the slice genus $g_4(K_n)$ under twistings, we prove that if $g(K_n) - g_4(K_n) < C$ for some constant $C$ for infinitely many integers $n > 0$ or $g(K_n) / g_4(K_n) \\to 1$ as $n \\to \\infty$, then either the winding number of $K$ about $c$ is zero or the winding number equals the wrapping number. 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