Surgeries and tangle replacement in alternating diagrams

Surgeries and tangle replacement in alternating diagrams

-
Duncan Mccoy , University of Glasgow
Fine Hall 314

The question of when the double branched cover of a knot K can be constructed by surgery on a knot in S^3 arises when one considers whether K can be obtained by rational tangle replacement on the unknot. I will explain how one can show that the double branched cover of an alternating knot or link can arise by non-integer surgery on a knot in S^3 if and only if it has an alternating diagram which can be obtained from an almost-alternating diagram of the unknot by an appropriate tangle replacement. As a special case, this shows that the alternating knots with unknotting number one are precisely those with an unknotting crossing in an alternating diagram.