Abstract
We compute the unknotting number for infinite families of knots by using a famous inequality due to Murasugi that relates the unknotting number of a knot to the signature of the same knot. Also, we determine the unknotting number and show it is equal to two for some knots in the knot table with twelve crossings or less by another inequality due to Nakanishi that relates the unknotting number of a knot to the surgerical description number of the knot and by a theorem that is due to Kanenobu and Murakami.
Highlights
IntroductionFor a classical knot K in S3, the unknotting number u(K) is the minimum number of crossing changes required to deform the knot K to the unknot where
For a classical knot K in S3, the unknotting number u(K) is the minimum number of crossing changes required to deform the knot K to the unknot whereReceived: March 29, 2015 §Correspondence author c 2015 Academic Publications, Ltd. url: www.acadpubl.euM
Even though some algebraic techniques are known to give a lower bound for u(k), the unknotting number for many knots in the knot table with twelve crossings or less are still undetermined
Summary
For a classical knot K in S3, the unknotting number u(K) is the minimum number of crossing changes required to deform the knot K to the unknot where. We use the following famous inequality due to Murasugi [8] that relates the unknotting number u(K) to the signature σ(K) of the knot K to compute the unknotting number of infinite families of knots. We use the following inequality due to Nakanishi [10, 11] that relates the unknotting number u(K) to the surgerical description number (K) of the knot. K to determine the unknotting number of the knots 937, 103 and 1020 and show it is equal to two. We use the following theorem to determine the unknotting number of the knots 12a802 and 12a1166 and show it is equal to two. K can be expressed as C(a, a1, a2, . . . , ak, ±2, −ak, . . . , −a2, −a1) in Conway’s normal form [4]
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