Seifert matrix ($A$)

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.