Witt Rings and Algebraic Knot Concordance
The (knot) concordance group was introduced by Fox and Milnor in 1966. Since then some progress has been made studying both slice knots and concordance, though even some basic questions remain unanswered. We discuss the construction of the concordance group starting from knotted circles in S^3. Of the two related theories of concordance, we focus on the smooth setting rather than the topological setting.In 1969, Levine classified higher dimensional analogues of concordance and provided great insight into the classical version mentioned above. His scheme allows tools developed in the purely algebraic setting of the theory of symmetric bilinear forms (specifically, Witt theory) to be applied to questions of concordance through use of an algebraic concordance group. We develop this notion of algebraic concordance alongside the corresponding notions from Witt theory.
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