Knot signature functions are independent

Jae Choon Cha; Charles Livingston

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Knot signature functions are independent

Jae Choon Cha·Charles Livingston

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Abstract

To each unit complex number with positive imaginary part there is defined a Tristram-Levine knot signature function. The set of all such signature functions is linearly independent as a set of functions defined on the set of all knots. The set of averaged signature functions forms a linearly independent set of homomorophisms on the knot concordance group. However, for each unit root of an Alexander polynomial, there is a slice knot with nonvanishing signature at that root and its conjugate, and nowhere else. These results hold for knots in all odd dimension.

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Jae Choon Cha, Charles Livingston. 2003-01-26. Knot signature functions are independent. https://arxiv.org/abs/math/0208225

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