T3 4 is in fact the knot 819, which is the rst non-alternating knot in the tables. cylinder 7! twisted up 7! T3 4 The trefoil can be seen as T2 3 and the knot 51 as T2 5. The torus is irrelevant | one is only interested in the resulting link Tp q formed from the strands drawn on its surface | but it certainly helps in visualising the link. Torus links are produced bychoosing a pair of integers p>0 q, forming a cylinder with p strings running along it, twisting it up through \q=p full twists" (the sign of q determines the direction of twist) and gluing its ends together to form an unknotted torus in R 3. These can be useful as examples, counterexamples, tests of conjectures, and in connection with other topics. There are many ways of creating whole families of knots or links with similar properties. (Proving that the unknotting number of the trefoil is not zero is equivalent to proving it distinct from the unknot: proving that u(51) > 1iseven harder.) 1.6. In each case one may obtain an upper bound simply by exhibiting a diagram and a set of unknotting crossings, but the lower bound is much harder. In fact the trefoil has u =1 and the knot 51 has u =2. The unknot is clearly the only knot with unknotting number u =0. In other words, if one is allowed to let the string of the knot pass through itself, one can clearly reduce it to the unknot: the question is how many times one needs to let it cross itself in this way. The unknotting number u(K) of a knot K is the minimum, over all diagrams D of K, of the minimal number of crossing changes required to turn D into a diagram of the unknot. This means that given any knot diagram, it is possible to turn it into a diagram of the unknot simply by changing some of its crossings. In fact there is always a way of assigning the crossings so that the result is an unknot. If one repeats the \random knot" construction above but puts in the crossings so that the rst time one reaches any given crossing one goes over (one will eventually come back to it on the underpass), one produces mainly unknots. ![]() ![]() The rst non-alternating one is 819 in the tables. Is every alternating diagram minimal? In particular, does every non-trivial alternating diagram represent a non-trivial knot? The answer turns out to be (with a minor quali cation) yes, as we will prove with the aid of the Jones polynomial (this was only proved in 1985). In fact one may ask, as Tait did: Question 1.4.4. 6 JUSTIN ROBERTS If one carries this out it seems that the results \really are knotted".
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