Symmetrized Seifert Form in Topological Invariants

• Symmetrized Seifert form is a symmetric bilinear form obtained by adding a Seifert matrix to its transpose, reflecting linking properties of oriented Seifert surfaces.
• It uses block triangular structure of homogeneous links to factor determinants and compute link signatures and polynomial invariants with reduced complexity.
• This framework enables precise analysis of link invariants and minimal genus conditions by decomposing complex surfaces into manageable blocks.

The symmetrized Seifert form is a central object in low-dimensional topology, singularity theory, and the paper of links, knots, and surface embeddings. For a link with an oriented Seifert surface, the Seifert matrix encodes algebraic data from linking numbers of cycles and their positive normal push-offs, but is generically nonsymmetric. Symmetrization yields a genuine symmetric bilinear form on homology, providing a powerful tool for analyzing link invariants, singularity invariants, and embeddings. In the setting of homogeneous links, the algebraic structure of the Seifert matrix—particularly in its block triangular form—directly dictates the properties of its symmetrization and thus many associated topological invariants (1102.0890).

1. Seifert Matrix Construction and Symmetrization

Given an oriented link $L$ and a choice of oriented Seifert surface $F$, a basis $\{a_1, \ldots, a_n\}$ for $H_1(F)$ is fixed. The Seifert matrix $M$ is defined by

$a_{ij} = \mathrm{lk}(a_i, a_j^+),$

where $a_j^+$ denotes a push-off of $a_j$ in the positive normal direction to $F$.

While $M$ is a bilinear form on $H_1(F)$, it is generally not symmetric. The symmetrized Seifert form is given by

$S = M + M^T,$

that is, for cycles $x$ and $y$,

$$S(x, y) = \mathrm{lk}(x, y^+) + \mathrm{lk}(y, x^+).$$

This process of symmetrization produces a symmetric bilinear form on $H_1(F)$, providing a more direct reflection of many topological quantities, such as link signatures, than the original nonsymmetric Seifert form.

2. Block Triangular Structure for Homogeneous Links

When the Seifert surface arises from the Seifert algorithm applied to a diagram of a homogeneous link, the associated Seifert graph $G$ decomposes into blocks (maximal subgraphs without cut-vertices). By grouping the basis elements of $H_1(F)$ according to these blocks, the Seifert matrix $M$ admits an upper block triangular form: $$M = \begin{bmatrix} M_1 & 0 & \cdots & 0 \ * & M_2 & \ddots & \vdots \ \vdots & \ddots & \ddots & 0 \ * & \cdots & * & M_k \end{bmatrix},$$ where each $M_i$ is the Seifert matrix associated to the block $B_i$ of the Seifert graph.

Key features of this form:

• The determinant of $M$ factors as $\det M = \prod_{i=1}^k \det M_i$.
• Each $M_i$ is associated with a "geometric piece" of $F$, and for homogeneous links all crossings in each block have the same sign $\epsilon_i$.
• For each block, classical results ensure $\det M_i \neq 0$ and $$\operatorname{sign}(\det M_i) = (-\epsilon_i)^{r_i}$$, where $r_i$ is the rank of the block.

The block triangular form is essential in analyzing the algebraic structure of the Seifert matrix and its symmetrization, and in reducing complex invariants to computations on smaller, more manageable building blocks of the Seifert surface.

3. Symmetrized Seifert Form in Block Structure

Symmetrization interacts with the block structure as follows. If off-diagonal lower blocks $A_{ij}$ may be nonzero, then

$$M = \begin{bmatrix} M_1 & 0 & \cdots & 0 \ A_{21} & M_2 & \ddots & \vdots \ \vdots & \ddots & \ddots & 0 \ A_{k1} & \cdots & A_{k,k-1} & M_k \end{bmatrix},$$

so,

$$S = M + M^T = \begin{bmatrix} M_1+M_1^T & A_{21}^T & \cdots & A_{k1}^T \ A_{21} & M_2+M_2^T & \ddots & \vdots \ \vdots & \ddots & \ddots & A_{k,k-1}^T \ A_{k1} & \cdots & A_{k,k-1} & M_k+M_k^T \end{bmatrix}.$$

While $S$ is generally not block diagonal, the key point is that each diagonal block $S_i = M_i+M_i^T$ is a nondegenerate symmetric bilinear form. The symmetrized form $S$ thus acquires a structure in which its nondegenerate pieces correspond to contributions from the individual blocks of the Seifert graph.

This structure ensures that invariants computed from $S$, such as link signatures or leading coefficients of Alexander/Conway polynomials, can be traced to the behavior of the individual blocks.

4. Determinant Factorization and Topological Invariants

The nonvanishing of each $\det M_i$ is essential:

• Each $M_i$ corresponds to a geometric region $F_i$ in the Seifert surface, and nondegeneracy implies substantial linking/intersection among the cycles of that region.
• The block triangular form gives

$\det M = \prod_{i=1}^k \det M_i,$

and for homogeneous links, each

$\det M_i = (-\epsilon_i)^{r_i} |\det M_i|.$

• Thus, the Conway polynomial's leading coefficient or the Alexander polynomial's leading term can be written as a product over the blocks:

$$c = (-1)^n \det M = \prod_{i=1}^k \epsilon_i^{r_i} |\det M_i|.$$

The fact that $S_i = M_i+M_i^T$ is nondegenerate for each block secures the non-degeneracy of the symmetrized Seifert form and underlies the calculation of many link invariants.

5. Significance for Link Invariants and Minimal Genus

The nondegeneracy and block structure of the symmetrized Seifert form have concrete implications:

• The signature of $S$ (and thus the Murasugi–Tristram signature invariants) is determined by the signatures of the $S_i$.
• For homogeneous links, block structure analysis leads directly to the minimal genus property of the Seifert surface constructed by the Seifert algorithm.
• The explicit determinant factorization simplifies computation and provides theoretical understanding of how contributions from individual geometric blocks assemble to the overall invariant.

Consequently, the analysis of $S$ via block structure systematically reduces the complexity of both numerical and algebraic computations. It also clarifies how local geometric features (e.g., the sign and configuration of crossings in each block) determine global invariants.

6. Summary of Relevant Formulas

The following table summarizes key constructions:

Construction	Formula / Definition	Role
Seifert matrix	$a_{ij} = \mathrm{lk}(a_i, a_j^+)$	Encodes linking/push-off in $F$
Symmetrization	$S = M + M^T$	Symmetric bilinear form on $H_1(F)$
Block triangular	$M = [M_1, 0; *, M_2;\ldots; *,\ldots,M_k]$	Organization by Seifert graph blocks
Determinant factor	$\det M = \prod_{i=1}^k \det M_i$	Simplifies computation of Alexander/Conway polynomials
Diagonal block sign	$$\operatorname{sign}(\det M_i) = (-\epsilon_i)^{r_i}$$	Sign control per block for homogeneous links
Conway leading coeff	$$c = (-1)^n \det M = \prod_{i=1}^k \epsilon_i^{r_i} |\det M_i|$$	Links algebraic and genus-minimality properties

These formulas codify the precise way in which the structure of the Seifert matrix, its block decomposition, and its symmetrization together determine key topological and algebraic invariants of homogeneous links.

7. Implications for Further Research and Constructions

The methodology described for block triangular forms and their symmetrized analogues forms the backbone of many modern investigations in knot theory and 3-manifold topology:

• The analysis of symmetrized Seifert forms is critical in the classification of concordance classes, signature invariants, and the computation of polynomial link invariants.
• The block decomposition approach provides a clear strategy for decomposing complex link diagrams into elementary constituents, each contributing additively or multiplicatively to global invariants.
• Extensions of these techniques appear naturally in the paper of link cobordism, S-equivalence of Seifert forms, and the algebraic classification of links via their associated forms.

For homogeneous links, this approach is particularly powerful due to the alignment of geometric and algebraic features, enabling precise control over invariants and facilitating proofs regarding minimal genus surfaces and determinant computations (1102.0890).