# 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 ${\displaystyle V+V^{\perp }}$, 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.

## 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)