Exercise Set VII MATHS 441
Due: November 2, 2009 Dr. Fischer
1. Undo the knot in Figure 1 of the attachment. That is, find a sequence of
Reidemeister moves that transform the shown diagram into a diagram of the
unknot.
2. Prove that tricolorability is a link invariant.
3. Let K1and K2be two (oriented) knots. Suppose that K1is tricolorable.
Prove that K1#K2is tricolorable.
4. Prove that there is no knot Ksuch that 31#Kis equivalent to the unknot.
5. Consider the three links shown in Figure 2 of the attachment.
(a) For each link, determine whether or not it is tricolorable.
(b) For each link, determine the absolute value of the linking number.
(c) Prove that no two of these three links are equivalent.
6. Suppose that Dis a diagram of a knot K.
(a) Prove that one can always change Dinto a diagram of the unknot by
changing some of the crossings from overpasses to underpasses or vice
versa.
(b) The minimum number of crossing changes required to change some dia-
gram of Kinto a diagram of the unknot is called the unknotting number
of K, denoted by u(K). Consider the knot 83shown in Figure 3 of the
attachment. Prove that u(83)62.
(c) Let c(K) denote the crossing number of K. Prove that u(K)6c(K)/2.
[Hints are attached!]
