Signature of a knot

The signature of a knot is a topological invariant in knot theory. It may be computed from the Seifert surface.

Given a knot K in the 3-sphere, it has a Seifert surface S whose boundary is K. The Seifert form of S is the pairing ${\displaystyle \phi :H_{1}(S)\times H_{1}(S)\to \mathbb {Z} }$ given by taking the linking number ${\displaystyle lk(a^{+},b^{-})}$ where ${\displaystyle a,b\in H_{1}(S)}$ and ${\displaystyle a^{+},b^{-}}$ indicate the translates of a and b respectively in the positive and negative directions of the normal bundle to S.

Given a basis ${\displaystyle b_{1},...,b_{2g}}$ for ${\displaystyle H_{1}(S)}$ (where g is the genus of the surface) the Seifert form can be represented as a 2g-by-2g Seifert matrix V, ${\displaystyle V_{ij}=\phi (b_{i},b_{j})}$. The signature of the matrix ${\displaystyle V+V^{t}}$, thought of as a symmetric bilinear form, is the signature of the knot K.

Slice knots are known to have zero signature.

The Alexander module formulation

Knot signatures can also be defined in terms of the Alexander module of the knot complement. Let ${\displaystyle X}$ be the universal abelian cover of the knot complement. Consider the Alexander module to be the first homology group of the universal abelian cover of the knot complement: ${\displaystyle H_{1}(X;\mathbb {Q} )}$. Given a ${\displaystyle \mathbb {Q} [\mathbb {Z} ]}$-module ${\displaystyle V}$, let ${\displaystyle {\overline {V}}}$ denote the ${\displaystyle \mathbb {Q} [\mathbb {Z} ]}$-module whose underlying ${\displaystyle \mathbb {Q} }$-module is ${\displaystyle V}$ but where ${\displaystyle \mathbb {Z} }$ acts by the inverse covering transformation. Blanchfield's formulation of Poincaré duality for ${\displaystyle X}$ gives a canonical isomorphism ${\displaystyle H_{1}(X;\mathbb {Q} )\simeq {\overline {H^{2}(X;\mathbb {Q} )}}}$ where ${\displaystyle H^{2}(X;\mathbb {Q} )}$ denotes the 2nd cohomology group of ${\displaystyle X}$ with compact supports and coefficients in ${\displaystyle \mathbb {Q} }$. The universal coefficient theorem for ${\displaystyle H^{2}(X;\mathbb {Q} )}$ gives a canonical isomorphism with ${\displaystyle Ext_{\mathbb {Q} [\mathbb {Z} ]}(H_{1}(X;\mathbb {Q} ),\mathbb {Q} [\mathbb {Z} ])}$ (because the Alexander module is ${\displaystyle \mathbb {Q} [\mathbb {Z} ]}$-torsion). Moreover, just like in the quadratic form formulation of Poincaré duality, there is a canonical isomorphism of ${\displaystyle \mathbb {Q} [\mathbb {Z} ]}$-modules ${\displaystyle Ext_{\mathbb {Q} [\mathbb {Z} ]}(H_{1}(X;\mathbb {Q} ),\mathbb {Q} [\mathbb {Z} ])\simeq Hom_{\mathbb {Q} [\mathbb {Z} ]}(H_{1}(X;\mathbb {Q} ),[\mathbb {Q} [\mathbb {Z} ]]/\mathbb {Q} [\mathbb {Z} ])}$, where ${\displaystyle [\mathbb {Q} [\mathbb {Z} ]]}$ denotes the field of fractions of ${\displaystyle \mathbb {Q} [\mathbb {Z} ]}$. This isomorphism can be thought of as a sesquilinear duality pairing ${\displaystyle H_{1}(X;\mathbb {Q} )\times H_{1}(X;\mathbb {Q} )\to [\mathbb {Q} [\mathbb {Z} ]]/\mathbb {Q} [\mathbb {Z} ]}$ where ${\displaystyle [\mathbb {Q} [\mathbb {Z} ]]}$ denotes the field of fractions of ${\displaystyle \mathbb {Q} [\mathbb {Z} ]}$. This form takes value in the rational polynomials whose denominators are the Alexander polynomial of the knot, which as a ${\displaystyle \mathbb {Q} [\mathbb {Z} ]}$-module is isomorphic to ${\displaystyle \mathbb {Q} [\mathbb {Z} ]/\Delta K}$. Let ${\displaystyle tr:\mathbb {Q} [\mathbb {Z} ]/\Delta K\to \mathbb {Q} }$ be any linear function which is invariant under the involution ${\displaystyle t\longmapsto t^{-1}}$, then composing it with the sesquilinear duality pairing gives a symmetric bilinear form on ${\displaystyle H_{1}(X;\mathbb {Q} )}$ whose signature is an invariant of the knot.

All such signatures are concordance invariants, so all signatures of slice knots are zero. The sesquilinear duality pairing respects the prime-power decomposition of ${\displaystyle H_{1}(X;\mathbb {Q} )}$—i.e.: the prime power decomposition gives an orthogonal decomposition of ${\displaystyle H_{1}(X;\mathbb {R} )}$. Cherry Kearton has shown how to compute the Milnor signature invariants from this pairing, which are equivalent to the Tristram-Levine invariant.

See also

• Link concordance

References

• C.Gordon, Some aspects of classical knot theory. Springer Lecture Notes in Mathematics 685. Proceedings Plans-sur-Bex Switzerland 1977.
• J.Hillman, Algebraic invariants of links. Series on Knots and everything. Vol 32. World Scientific.
• C.Kearton, Signatures of knots and the free differential calculus, Quart. J. Math. Oxford (2), 30 (1979).
• J.Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44, 229-244 (1969)
• J.Milnor, Infinite cyclic coverings, J.G. Hocking, ed. Conf. on the Topology of Manifolds, Prindle, Weber and Schmidt, Boston, Mass, 1968 pp. 115–133.
• K.Murasugi, On a certain numerical invariant of link types, Trans. Amer. Math. Soc. 117, 387-482 (1965)
• A.Ranicki On signatures of knots Slides of lecture given in Durham on 20 June 2010.
• H.Trotter, Homology of group systems with applications to knot theory, Ann. of Math. (2) 76, 464-498 (1962)