Papers โบ Mock Seifert matrices and unoriented algebraic concordance
Mock Seifert matrices and unoriented algebraic concordance
Hans U. Boden, Homayun Karimi
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A mock Seifert matrix is an integral square matrix representing the Gordon-Litherland form of a pair (K,F), where K is a knot in a thickened surface and F is an unoriented spanning surface for K. Using these matrices, we introduce a new notion of unoriented algebraic concordance, as well as a new group denoted ๐ ๐ข^โค and called the unoriented algebraic concordance group. This group is abelian and infinitely generated. There is a surjection ฮป๐ ๐ โ๐ ๐ข^โค, where ๐ ๐ denotes the virtual knot concordance group. Mock Seifert matrices can also be used to define new invariants, such as the mock Alexander polynomial and mock Levine-Tristram signatures. These invariants are applied to questions about virtual knot concordance, crosscap numbers, and Seifert genus for knots in thickened surfaces. For example, we show that ๐ ๐ข^โค contains a copy of โค^โ โ(โค/2)^โ โ(โค/4)^โ.
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