Type: Article
Publication Date: 1976-01-01
Citations: 83
DOI: https://doi.org/10.1090/s0002-9947-1976-0388373-0
Let <italic>L</italic> be an oriented tame link in the three sphere <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S cubed"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{S^3}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We study the Murasugi signature, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma left-parenthesis upper L right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>σ<!-- σ --></mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>L</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\sigma (L)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and the nullity, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="eta left-parenthesis upper L right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>η<!-- η --></mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>L</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\eta (L)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. It is shown that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma left-parenthesis upper L right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>σ<!-- σ --></mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>L</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\sigma (L)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a locally flat topological concordance invariant and that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="eta left-parenthesis upper L right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>η<!-- η --></mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>L</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\eta (L)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a topological concordance invariant (no local flatness assumption here). Known results about the signature are re-proved (in some cases generalized) using branched coverings.