Divisor (algebraic geometry)

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 $\varphi :H_1(S)\times H_1(S)\to \mathbb{\mathbb{Z}}$ given by taking the linking number $lk(a^{+},b^{-})$ where $a,b\in H_1(S)$ and $a^{+},b^{-}$ indicate the translates of a and b respectively in the positive and negative directions of the normal bundle to S.

Given a basis $b_1,...,b_{2g}$ for $H_1(S)$ (where g is the genus of the surface) the Seifert form can be represented as a 2g-by-2g Seifert matrix V, $V_{ij}=\varphi (b_i,b_j)$. The signature of the matrix $V+V^{\perp }$, 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 $X$ 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: $H_1(X;\mathbb{\mathbb{Q}})$. Given a $\mathbb{\mathbb{Q}}[\mathbb{\mathbb{Z}}]$-module $V$, let $\bar{V}$ denote the $\mathbb{\mathbb{Q}}[\mathbb{\mathbb{Z}}]$-module whose underlying $\mathbb{\mathbb{Q}}$-module is $V$ but where $\mathbb{\mathbb{Z}}$ acts by the inverse covering transformation. Blanchfield's formulation of Poincaré duality for $X$ gives a canonical isomorphism $H_1(X;\mathbb{\mathbb{Q}})\simeq \overline{H^2(X;\mathbb{\mathbb{Q}})}$ where $H^2(X;\mathbb{\mathbb{Q}})$ denotes the 2nd cohomology group of $X$ with compact supports and coefficients in $\mathbb{\mathbb{Q}}$. The universal coefficient theorem for $H^2(X;\mathbb{\mathbb{Q}})$ gives a canonical isomorphism with $Ex{t}_{\mathbb{\mathbb{Q}}[\mathbb{\mathbb{Z}}]}(H_1(X;\mathbb{\mathbb{Q}}),\mathbb{\mathbb{Q}}[\mathbb{\mathbb{Z}}])$ (because the Alexander module is $\mathbb{\mathbb{Q}}[\mathbb{\mathbb{Z}}]$-torsion). Moreover, just like in the quadratic form formulation of Poincaré duality, there is a canonical isomorphism of $\mathbb{\mathbb{Q}}[\mathbb{\mathbb{Z}}]$-modules $Ex{t}_{\mathbb{\mathbb{Q}}[\mathbb{\mathbb{Z}}]}(H_1(X;\mathbb{\mathbb{Q}}),\mathbb{\mathbb{Q}}[\mathbb{\mathbb{Z}}])\simeq Ho{m}_{\mathbb{\mathbb{Q}}[\mathbb{\mathbb{Z}}]}(H_1(X;\mathbb{\mathbb{Q}}),[\mathbb{\mathbb{Q}}[\mathbb{\mathbb{Z}}]]/\mathbb{\mathbb{Q}}[\mathbb{\mathbb{Z}}])$, where $[\mathbb{\mathbb{Q}}[\mathbb{\mathbb{Z}}]]$ denotes the field of fractions of $\mathbb{\mathbb{Q}}[\mathbb{\mathbb{Z}}]$. This isomorphism can be thought of as a sesquilinear duality pairing $H_1(X;\mathbb{\mathbb{Q}})\times H_1(X;\mathbb{\mathbb{Q}})\to [\mathbb{\mathbb{Q}}[\mathbb{\mathbb{Z}}]]/\mathbb{\mathbb{Q}}[\mathbb{\mathbb{Z}}]$ where $[\mathbb{\mathbb{Q}}[\mathbb{\mathbb{Z}}]]$ denotes the field of fractions of $\mathbb{\mathbb{Q}}[\mathbb{\mathbb{Z}}]$. This form takes value in the rational polynomials whose denominators are the Alexander polynomial of the knot, which as a $\mathbb{\mathbb{Q}}[\mathbb{\mathbb{Z}}]$-module is isomorphic to $\mathbb{\mathbb{Q}}[\mathbb{\mathbb{Z}}]/\mathrm{\Delta }K$. Let $tr:\mathbb{\mathbb{Q}}[\mathbb{\mathbb{Z}}]/\mathrm{\Delta }K\to \mathbb{\mathbb{Q}}$ be any linear function which is invariant under the involution $t⟼t^{-1}$, then composing it with the sesquilinear duality pairing gives a symmetric bilinear form on $H_1(X;\mathbb{\mathbb{Q}})$ 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 $H_1(X;\mathbb{\mathbb{Q}})$ -- ie: the prime power decomposition gives an orthogonal decomposition of $H_1(X;\mathbb{ℝ})$. 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)

See also

• Link concordance

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