| This thesis consists of two parts, the first part is in the setting of algebraic knot theory while the second studies ideas in representation theory.;In the first part of this work, we study generalizations of a classical link invariant--the multivariable Alexander polynomial--to tangles. The starting point is Archibald's tMVA invariant for virtual tangles which lives in the setting of circuit algebras. Using the Hodge star map and restricting to tangles without closed components, we define a reduction of the tMVA to an invariant (rMVA) which is valued in matrices with entries equal to certain Laurent polynomials. We show the rMVA has the structure of a metamonoid morphism and is further equivalent to a tangle invariant defined by Bar-Natan. This invariant also reduces to the Gassner representation on braids and has a partially defined trace operation for closing open strands of a tangle.;In the second part, we look at crystals and the cactus group. The crystals for a finite-dimensional complex reductive Lie algebra g encode the structure of its representations, yet can also reveal surprising new structure of their own. In this work, we construct a group J g, the "cactus group'', using the Dynkin diagram of g and show that it acts combinatorially on any g-crystal via the Schutzenberger involutions. Henriques and Kamnitzer studied Jn = Jg ln, and constructed an action of it on n-tensor products of g-crystals, for any g as above. We discuss the crystal corresponding to the gln x glm-representation Lambda N(Cn ⊗ Cm), derive skew Howe duality on the crystal level and show that the two cactus group actions agree in this setting. An application of this result is discussed in studying a family of maximal commutative subalgebras of the universal enveloping algebra, the shift of argument and Gaudin algebras, where an algebraically constructed monodromy action is expected to match that of the cactus group. |
