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

.

Seifert genus ( $g_{s} (K)$ )

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

Seifert form ( $θ_{F}$ )

For oriented curves

x, y

on a Seifert surface, the Seifert form is

θ_{F} ([x], [y]) = lk (x^{+}, y),

where

x^{+}

is a positive normal push-off of

x

.

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.

Alexander polynomial of a knot ( $Δ_{K} (t)$ )

Up to multiplication by a unit

\pm t^{n}

, the Alexander polynomial is

Δ_{K} (t) = det (t A - A^{T}) .

It satisfies

Δ_{K} (t^{- 1}) ≐ Δ_{K} (t)

and

Δ_{K} (1) = \pm 1

.

Knot determinant ( $det K$ )

The knot determinant is

∣ Δ_{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

.

Levine-Tristram signature ( $σ_{ω} (K)$ )

For

ω \in S^{1} ∖ {1}

, the Levine-Tristram signature is the signature of the Hermitian matrix

H_{ω} = (1 - ω) A + (1 - \overline{ω}) A^{T} .

It is locally constant away from unit roots of the Alexander polynomial of a knot.

Algebraic concordance of knots

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

Metabolic Seifert form

A nonsingular Seifert form on a

2 g

-dimensional space is metabolic when it vanishes on a

g

-dimensional subspace called a metabolizer.

Algebraic concordance group over a field ( $A C_{F}$ )

The algebraic concordance group over a field

F

is the Witt group of nonsingular Seifert forms over

F

, modulo metabolic forms.

Isometric structure ( $(V, Q, T)$ )

An isometric structure consists of a finite-dimensional vector space

V

, a nonsingular symmetric bilinear form

Q

, and a

Q

T

with the required nondegeneracy at

\pm 1

. Metabolic isometric structures are quotiented out to form

W_{F}

.

Witt group of isometric structures ( $W_{F}$ )

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

Primary component of an isometric structure ( $V_{δ}$ )

For an irreducible symmetric Laurent polynomial

δ

, the primary component is

V_{δ} = ker δ (T)^{N}

for sufficiently large

N

. Distinct symmetric primary components are orthogonal, so restriction defines a projection

W_{F} \to W_{F}^{δ}

.

P-adic algebraic-concordance obstruction

Extending an algebraic-concordance class from

Q

to a p-adic field

Q_{p}

detects torsion invisible over the real numbers. For

p \equiv 3 (mod 4)

, an odd-dimensional second-residue form generates the order-four part of the local Witt group.

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Seifert surface by

Wikipedia Bot  1

A Seifert surface is a surface used in the field of topology, particularly in the study of knots and links in three-dimensional space. Named after Herbert Seifert, these surfaces are oriented surfaces that are bounded by a given link in the three-dimensional sphere \( S^3 \). The key properties and characteristics of Seifert surfaces include: 1. **Boundary**: The boundary of a Seifert surface is a link in \( S^3 \).

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