Consider Unsolved Question 2 on p.155 of Adams, where a formula is given for the Jones polynomial of an (m, n)-torus knot. (a) Describe what torus knots are, giving references and any plots, diagrams etc. you might think useful. In particular you should include some plots generated with the Mathematica files provided. (b) Find m, n such that the trefoil is an (m,n)-torus knot, and hence calculate the Jones polynomial of the trefoil using the formula given in Unsolved Question 2. Compare with your solution to Q.1 above. (c) Use the formula to find the Jones polynomial of the knot with 5 crossings which is a torus knot (justify your choice), and compare with the Jones polynomial found using Mathematica. Unsolved Question 2 Vaughan Jones has proved that the Jones polynomial of an (m, n)-torus knot is t(m-1)(n-1)/2(1 - tm+1 – {n+1 + tm+n)/(1 - 12). The only known proof, however, relies on algebras and is relatively difficult. Find a simple proof of this fact. (Maybe relate the Jones polynomial of an (m, n)-torus knot to the Jones polynomial of a simpler torus knot.) The Jones polynomial can be shown to satisfy a skein relation of its own. Let L+, L_, and Lo be three oriented link projections that are identical except where they appear as in Figure 6.13. X X ) x > L. L L. Figure 6.13 Three link projections that are almost identical.