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Abstract (summary): This thesis consists of two parts. In the main part of the thesis we introduce an extension of the Alexander polynomial to tangles, known as Γ-calculus or Gassner calculus, which has appeared in [BNS13, Hal16] and various talks by Prof Bar-Natan. Our main object of study is w-tangles, which we describe using the language of meta-monoids (see [BNS13, BN15a, Hal16]). There is a map from usual tangles to w-tangles and so an invariant of w-tangles induces an invariant of usual tangles. Using the language of Γ-calculus, we rederive certain important properties of the Alexander polynomial, most notably the Fox-Milnor condition on the Alexander polynomials of ribbon knots [Lic97, FM66]. We argue that our proof has some potential for generalization which may help tackle the slice-ribbon conjecture. In a sense this thesis is an extension of [BNS13]. In the second part of the thesis, we study the associated graded space of w-tangles, which is the space of arrow diagrams [BND16, BND14]. We describe an expansion of w-tangles, i.e. a map from w-tangles to its associated graded space. The concept of expansions is inspired by the Taylor expansions, and w-tangles have a much simpler expansion than usual tangles (for usual tangles an expansion is given by the Kontsevich integral [Oht02]). There is a relationship between arrow diagrams and Lie algebras. Using the expansion of w-tangles we recover Γ-calculus by choosing a particular Lie algebra, namely the two-dimensional non-abelian Lie algebra. We give a commutative diagram that summarizes the spaces and maps involved. The second part of the thesis is more or less independent of the first part.