By E. M. Friedlander, M. R. Stein

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13) Find a sequence of Markov moves that transforms the closure of the braid σ12 σ23 σ14 σ2 into the closure of the braid σ12 σ2 σ14 σ23 . ∆= (14) Garside’s fundamental braid ∆ ∈ Bn is defined as ∆ := (σ1 σ2 . . σn−1 )(σ1 σ2 . . σn−2 ) . . (σ1 σ2 )(σ1 ) . (a) Prove that σi ∆ = ∆σn−i for every standard generator σi ∈ Bn . (b) Prove that ∆2 = (σ1 σ2 . . σn−1 )n . (c) Check that ∆2 belongs to the centre Z(Bn ) of the braid group. (d) Show that any braid can be represented as a product of a certain power (possibly negative) of ∆ and a positive braid, that is, a braid that contains only positive powers of standard generators σi .

It can be obtained as a combination of the first version, ε , −ε with the moves V Ω1 and V Ω3 . Polyak [Po2]) Show that the following moves V Ω1 : V Ω↑↓ 2 : ε ε −ε −ε V Ω+++ : 3 are sufficient to generate all Reidemeister moves V Ω1 , V Ω2 , V Ω3 . Chapter 2 Knot invariants Knot invariants are functions of knots that do not change under isotopies. The study of knot invariants is at the core of knot theory; indeed, the isotopy class of a knot is, tautologically, a knot invariant. 1. Definition and first examples Let K be the set of all equivalence classes of knots.

A) t−2 J (b) tJ (c) t2 J + t2 J = (t + t−1 )J + t−1 J ; = (t + t−1 )J + t−2 J = t−2 J ; + t2 J . Compare these relations with those of Exercise 8 for the Conway polynomial. (18) Prove that for a link L with an odd number of components, J(L) is a polynomial in t and t−1 , and for a link L with an even number of components J(L) = t1/2 · (a polynomial in t and t−1 ). (19) Prove that for a link L with k components J(L) particular, J(K) t=1 = 1 for a knot K. (20) Prove that d(J(K)) dt =0 t=1 = (−2)k−1 .