Alternating knots with unknotting number one

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• 作者：Duncan McCoy ^{d.mccoy.1@research.gla.ac.uk" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Unknotting number ; Knots ; Dehn surgery ; Branched double covers
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：10 January 2017
• 年：2017
• 卷：305
• 期：Complete
• 页码：757-802
• 全文大小：829 K

文摘

We show that an alternating knot with unknotting number one has an unknotting crossing in any alternating diagram. We also prove that an alternating knot has unknotting number one if and only if its branched double cover arises as half-integer surgery on a knot in S³msup>S3msup>, thus establishing a converse to the Montesinos trick. Along the way, we reprove a characterisation of almost-alternating diagrams of the unknot originally due to Tsukamoto. These results are established using the obstruction to unknotting number one developed by Greene.