Knot Polynomial Invariants And Three-dimensional Manifolds Quantum Invariants

The body of this thesis is composed of three chapters. Chapter 1 The trip matrixes for linksL.Zulli [24] first constructed a matrix (mod 2) for computing the Kauffman bracket polynomials of knots, which is called trip matrix. We studied the trip matrixes of links here. The main result isTheorem 3.1 If the state S is obtained from the state AA---A by toggling the labels in positions i1, i2, ..., ip. Let Ts be the matrix obtained from the trip matrix by toggling the entries in the corresponding positions along the diagonal of the sub matrix in the up-left corner of T. Then# (L/S)=n+m-rank(Ts)chapter 2 Notes on polynomial invariants of K(A,B) and K.(P1, P2, ..., Pn)T.Kanenobu studied the structure of polynomial invariants of K (a, b) and K (p1,p2,...,pn )> where a, b',p1, p2, ...,pn are all number tangles. We studied the properties of the polynomial invariants of K (A, B) and K (P1,P2,..., Pn), where A,B; P1, P2,..., Pn are general tangles. The main results areProposition 2.1.2 In unorient caseProposition 2.1.3 In orient case K(A+2n, B-2n) is skein equivalent to K(A,B) for n Z . Proposition 2.2.2 In unarient case, if l, + 2 +...+ n = 0, i Z , (i = 1,2, ...,n), thenProposition 2.2.3 In orient caseTheorem 2.2.1 The statements on K (A,B) here are still valid for a sequence of Km (A, B) (m Z) .(see Fig. 6 in chapter 2)Chapter 3 Modular CategoryThis chapter consists of four sections. The most contents of the first three sections are taken from [23] .The section 4 is devote to a similar invariant of 3manifolds given by the author. The main result isTheorem 4.2 C(ML) = D-(c-1)(n-b1(ML)) (Mc*L ) is a topological invariant of ML, where n is thenumber of components of L, b1 (ML ) is the first Betti number of ML, c*L is the link consisting of c parallel copies of L.