L.H. Kauffman. Formal Knot Theory. Lecture Notes #115. Princeton University Press (1983).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of that theory (a graph is 1dimensional complex) However in this combinatorial context the theory of NZF has little to do 23 with very general approaches of algebraic topology. But these connections certainly influenced research on both sides (for example in the theory of invariants of knots [26]; compare also [27, 30] However what makes the theory of flows and tensions also interesting is that this theory unified and explained connections and gave easy proofs of several results which were regarded as isolated and peculiar. Let us give some examples: Proposition 6 If Phi is a ZZ 2 ....

L. H. Kauffman. Formal knot theory. Mathematical Notes, 30. Princeton, New Jersey: Princeton University Press, (1983).

....projection of a knot to perform calculations. For example, the Jones [18] and AlexanderConway [16] polynomials can be computed by a skein relation. Vassiliev [28] invariants obey similar skein relations [2] for singular knots. The combinatorics contained in the projection are used extensively in [22] and [23] for example, and the algebraic information that is encoded in either a diagram or a projection can be found in [3] In this paper, we explore projections of knotted surfaces in 4 space and their isotopies. By a knotted surface we mean smoothly embedded surface in 4 dimensional Euclidean ....

L. H. Kauffman, Formal Knot Theory, Princeton U. Press, Mathematical Notes # 30, 1983. 27

....2.3. If n is even, an arbitrary quadruple of (n; n 2) knots T = K 1 ;K 2 ;X 1 ; X 2 ) is realizable. In order to prove Theorem 2.1, we introduce a new knotting operation for high dimensional knots, high dimensional pass moves. The 1 dimensional case of Definition 2.1 is discussed on p. 146 of [K1]. Definition. Let (2k 1) knot K be defined by a smooth embedding g :6 2k 1 , S 2k 3 , where 6 2k 1 is PL homeomorphic to the standard (2k 1) sphere. k = 0: Let D k 1 x =f(x 1 ; x k 1 )j 6x 2 i 1g and D k 1 y =f(y 1 ; y k 1 )j 6y 2 i 1g. Let D k 1 x (r) f(x 1 ; ....

....conditions areequivalent. k = 0. 1) There exists a (2k 1) knot K 3 which is pass move equivalent to K 1 and cobordant to K 2 . 2) K 1 and K 2 satisfy the condition that ae Arf(K 1 ) Arf(K 2 ) when k is even oe(K 1 ) oe(K 2 ) when k is odd. The case k = 0 of Theorem 2. 5 follows from [K1], K2] x3 We discuss the case when three spheres intersect in a sphere. Let F i be closed surfaces (i = 1; 2; A surface (F 1 ;F 2 ; F ) link is a smooth submanifold L = K 1 ;K 2 ; K )ofS 4 , where K i is diffeomorphic to F i . If F i is orientable we assume that F i is ....

Kauffman, L., Formal knot theory, Mathematical Notes 30, Princeton University Press, 1983.

....2.3. If n is even, an arbitrary quadruple of (n; n 2) knots T = K 1 ; K 2 ; X 1 ; X 2 ) is realizable. In order to prove Theorem 2.1, we introduce a new knotting operation for high dimensional knots, high dimensional pass moves. The 1 dimensional case of Definition 2.1 is discussed on p. 146 of [K1]. Definition. Let (2k 1) knot K be defined by a smooth embedding g : Sigma 2k 1 , S 2k 3 , where Sigma 2k 1 is PL homeomorphic to the standard (2k 1) sphere. k = 0: Let D k 1 x =f(x 1 ; x k 1 )j Sigmax 2 i 1g and D k 1 y =f(y 1 ; y k 1 )j Sigmay 2 i 1g. ....

....are equivalent. k = 0. 1) There exists a (2k 1) knot K 3 which is pass move equivalent to K 1 and cobordant to K 2 . 2) K 1 and K 2 satisfy the condition that ae Arf(K 1 ) Arf(K 2 ) when k is even oe(K 1 ) oe(K 2 ) when k is odd. The case k = 0 of Theorem 2. 5 follows from [K1], K2] x3 We discuss the case when three spheres intersect in a sphere. Let F i be closed surfaces (i = 1; 2; A surface (F 1 ; F 2 ; F ) link is a smooth submanifold L = K 1 ; K 2 ; K ) of S 4 , where K i is diffeomorphic to F i . If F i is orientable we assume ....

Kauffman, L., Formal knot theory, Mathematical Notes 30, Princeton University Press, 1983.

....2.3. If n is even, an arbitrary quadruple of (n; n 2) knots T = K 1 ; K 2 ; X 1 ; X 2 ) is realizable. In order to prove Theorem 2.1, we introduce a new knotting operation for high dimensional knots, high dimensional pass moves. The 1 dimensional case of Definition 2.1 is discussed on p. 146 of [K1]. Definition. Let (2k 1) knot K be defined by a smooth embedding g : Sigma 2k 1 , S 2k 3 , where Sigma 2k 1 is PL homeomorphic to the standard (2k 1) sphere. k = 0: Let D k 1 x =f(x 1 ; x k 1 )j Sigmax 2 i 1g and D k 1 y =f(y 1 ; y k 1 )j Sigmay 2 i 1g. ....

....are equivalent. k = 0. 1) There exists a (2k 1) knot K 3 which is pass move equivalent to K 1 and cobordant to K 2 . 2) K 1 and K 2 satisfy the condition that ae Arf(K 1 ) Arf(K 2 ) when k is even oe(K 1 ) oe(K 2 ) when k is odd. The case k = 0 of Theorem 2. 5 follows from [K1], K2] x3 We discuss the case when three spheres intersect in a sphere. Let F i be closed surfaces (i = 1; 2; A surface (F 1 ; F 2 ; F ) link is a smooth submanifold L = K 1 ; K 2 ; K ) of S 4 , where K i is diffeomorphic to F i . If F i is orientable we assume ....

Kauffman, L., Formal knot theory, Mathematical Notes 30, Princeton University Press, 1983.

....By coloring diagrams with arbitrary t, we obtain a polynomial that generalizes the modulus. This polynomial is the Alexander polynomial. Alexander [AL] described it differently in his original paper, and there is a remarkable history to the development of this invariant. See [CF] FOX] CON] K1] [K2], K4] for more information. The flavor of this relationship can be seen by doing a little experiment in labeling the trefoil diagram shown in Figure 19. The circularity inherent in the knot diagram results in relations that must be satisfied by the module action. In Figure 19 we see directly by ....

....defining properties of the Jones polynomial, and later sections to the relationships with physics. Here are a set of axioms for the Jones polynomial. The polynomial was not discovered in the form of these axioms. The axioms are in a format analogous to the framework that John H. Conway [CON] K1] [K2], discovered for the Alexander polynomial. I am starting with these axioms because they give a quick access to the polynomial and to sample computations. Axioms for the Jones Polynomial 1. If two oriented links K and K are ambient isotopic, then VK(t) VK(t) 2. If U is an unknotted loop, then ....

L.H. Kauffman. Formal Knot Theory. Lecture Notes #115. Princeton University Press (1983).

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L.H. Kauffman, Formal knot theory, Mathematical Notes 30, Princeton University Press, 1983. 24

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