Seifert surface

A Seifert surface for an oriented knot $K$ is a compact connected oriented surface $F\subset S^3$ with oriented boundary $\partial F=K$.

A Hopf band is an annulus with one full twist whose boundary is a Hopf link. Plumbing it with another band adds a generator to the first homology of a Seifert surface. Its core has self-linking number $1$ or $-1$, depending on the twisting convention.

A Seifert hypersurface for an oriented codimension-two submanifold is an oriented codimension-one submanifold having it as boundary. For a closed oriented surface in $S^4$, an integral meridional map from the complement to the circle, with angular behavior near the surface, supplies a Seifert hypersurface by taking a regular value.

Apply the oriented smoothing to every crossing of an oriented knot diagram, fill the resulting Seifert circles by disjoint disks at suitable heights, and reconnect them by twisted bands at the original crossings. The boundary is the original knot. If the connected surface has $s$ disks and $c$ bands, its Euler characteristic is $s-c$ and its genus is $(1-s+c)/2$.

A Seifert circle is a component of the crossing-free diagram obtained by oriented smoothing at every crossing. These circles bound the disks used in the Seifert algorithm.

The Seifert genus is the smallest genus of a Seifert surface for the knot.

It minimizes the genus over all choices of Seifert surface.

The Seifert genus is additive under connected sum of knots:The upper bound comes from boundary-connected-summing minimal Seifert surfaces, while the reverse inequality follows by cutting any Seifert surface for the connected sum along a splitting sphere.

For oriented curves $x,y$ on a Seifert surface, the Seifert form iswhere $x^{+}$ is a positive normal push-off of $x$.

A Seifert matrix represents the Seifert form in an integral basis of $H_1(F)$. Its skew-symmetrization $A-A^T$ represents the intersection form of the surface and is unimodular.

Cut a knot exterior along a Seifert surface and stack copies to form its infinite cyclic cover. Linking-dual generators for the homology of the cut-open piece give the positive and negative inclusion matrices $V$ and $V^T$ in row convention. The gluing relations therefore have matrix $tV-V^T$. The H0 gluing map is multiplication by $t-1$ and is injective, so these relations present the entire Alexander module of a knot. Its order is the determinant, up to a Laurent unit.

Up to multiplication by a unit $\pm t^n$, the Alexander polynomial isIt satisfies ${\Delta }_K(t^{-1})\doteq {\Delta }_K(t)$ and ${\Delta }_K(1)=\pm 1$.

The Conway polynomial is normalized by ${\nabla }_U=1$ for the unknot, zero for a split unlink with more than one component, and the skein relation ${\nabla }_{L_{+}}-{\nabla }_{L_{-}}=z{\nabla }_{L_0}$. For a knot, the symmetric Alexander polynomial with value one at $t=1$ is ${\Delta }_K(t)={\nabla }_K(t^{1/2}-t^{-1/2})$. This fixes the Laurent-unit ambiguity of the Seifert-matrix presentation of the Alexander module.

The symmetric Alexander polynomial normalized by the Conway polynomial is uniquely fixed for a knot. Its sign matters in the signed Jones evaluation at minus one, whereas the knot determinant retains only the absolute value.

A minimal-genus Seifert surface has a $2g_s(K)$-by-$2g_s(K)$ Seifert matrix $V$. Each entry of $tV-V^T$ has degree at most one, so its determinant has breadth of a Laurent polynomial at most $2g_s(K)$. This lower bound on Seifert genus need not be sharp: an untwisted Whitehead double can have Alexander polynomial of a knot $1$ and positive Seifert genus.

Assume $g_s(K)>0$. A nontrivial connected sum of knots has positive genus summands, and additivity of Seifert genus applies. Its Alexander polynomial of a knot is the product of the summand polynomials. If the polynomial is irreducible, one summand has unit polynomial; the other must then have polynomial breadth $2g_s(K)$, exceeding twice its strictly smaller genus. This contradicts the Alexander breadth bound on Seifert genus. Irreducibility without the full-degree hypothesis alone does not imply primeness.

Every Laurent polynomial $f(t)\in \mathbb{Z}[t^{\pm 1}]$ satisfying $f(t^{-1})=f(t)$ and $f(1)=1$ is the Alexander polynomial of a knot up to multiplication by a unit $\pm t^k$.

The Alexander module is the first integral homology of the infinite cyclic cover of a knot exterior. Its deck transformation makes it a module over $\mathbb{Z}[t^{\pm 1}]$, and its first elementary ideal determines the Alexander polynomial of a knot.

Boundary-connected-summing Seifert surfaces gives a block-diagonal Seifert matrix, because basis curves in the separated summands have zero linking. The Seifert-matrix presentation of the Alexander module is block diagonal too. Thus the Alexander modules form a direct sum and their Alexander polynomials of a knot multiply up to a unit.

Given a presentation $⟨x_1,\dots ,x_n\mid r_1,\dots ,r_m⟩$ of a knot group, apply abelianization to the Fox derivatives $\partial r_j/\partial x_i$. The resulting matrix over $\mathbb{Z}[t^{\pm 1}]$ is an Alexander matrix; suitable maximal minors recover the Alexander polynomial of a knot.

For a reduced knot diagram with two adjacent starred regions, a Kauffman state chooses one corner at every crossing so that every unstarred region contains exactly one chosen corner. Terms in a determinant expansion of a Dehn-presentation Alexander matrix correspond bijectively to these states.

The knot determinant is $|{\Delta }_K(-1)|=|\det (A+A^T)|$. It is also the order of the first homology of the two-fold cover of $S^3$ branched over $K$.

For $\omega \in S^1\setminus \{1\}$, the Levine-Tristram signature is the signature of the Hermitian matrixIt is locally constant away from unit roots of the Alexander polynomial of a knot.

At $\omega =-1$, this becomes the signature of a knot.

The knot signature is the signature of the symmetrized Seifert matrix. Equivalently it is the signature of the double branched covering of $B^4$ over a pushed-in Seifert surface. At $\omega =-1$ it agrees with the Levine-Tristram signature. It is additive under connected sum of knots, changes sign under reflection, and obeys $|\sigma (K)|\le 2g_4(K)$. State the ambient orientation and push-off convention when assigning a sign.

For a reduced connected alternating knot diagram, its knot signature is $s_A(D)-c_{+}(D)-1$, where $s_A$ counts circles in the all-$A$ bracket smoothing state and $c_{+}$ counts positive crossings. The convention is that the $A$-smoothing at a positive braid crossing is the oriented smoothing, and a positive trefoil knot has knot signature $-2$. Reversing the global knot signature convention negates the formula.

Two Seifert forms are algebraically concordant when their difference is metabolic. Stable equivalence classes form the algebraic concordance group.

A nonsingular Seifert form on a $2g$-dimensional space is metabolic when it vanishes on a $g$-dimensional subspace called a metabolizer.

The metabolizer is a half-dimensional subspace on which the Seifert form restricts to zero.

The algebraic concordance group over a field $F$ is the Witt group of nonsingular Seifert forms over $F$, modulo metabolic forms.

An isometric structure consists of a finite-dimensional vector space $V$, a nonsingular symmetric bilinear form $Q$, and a $Q$-isometry $T$ with the required nondegeneracy at $\pm 1$. Metabolic isometric structures are quotiented out to form ${\mathcal{W}}_F$.

The Witt group of isometric structures identifies two isometric structures when their orthogonal difference is metabolic.

For an irreducible symmetric Laurent polynomial $\delta$, the primary component isfor sufficiently large $N$. Distinct symmetric primary components are orthogonal, so restriction defines a projection ${\mathcal{W}}_F\to {\mathcal{W}}_F^{\delta }$.

Extending an algebraic-concordance class from $\mathbb{Q}$ to a p-adic field ${\mathbb{Q}}_p$ detects torsion invisible over the real numbers. For $p\equiv 3(\mathrm{mod}4)$, an odd-dimensional second-residue form generates the order-four part of the local Witt group.