Knot

A knot is a smooth embedding $S^1\hookrightarrow S^3$, considered up to ambient isotopy.

An oriented knot is a knot together with a choice of orientation on its embedded circle.

The unknot is the knot bounding a smoothly embedded disk in $S^3$.

The connected sum removes a short unknotted arc from each of two oriented knots and joins the four endpoints by an orientation-compatible pair of arcs.

A composite knot is a connected sum of two nontrivial knots.

A prime knot is a nontrivial knot that is not a composite knot.

A splitting sphere meets a knot transversely in two points. It is trivial when one of the resulting one-string tangles is boundary-parallel in its three-ball; otherwise it exhibits a nontrivial connected-sum decomposition.

For coprime integers $p,q$, the torus knot $T_{p,q}$ winds $p$ times meridionally and $q$ times longitudinally on an unknotted torus in $S^3$.

A satellite knot $P(K)$ is obtained by embedding a patterned solid torus into a tubular neighborhood of a companion knot $K$.

A pattern is a knot $P$ inside a solid torus. Embedding that torus as a tubular neighborhood of $K$ produces the satellite $P(K)$.

The winding number is the integer represented by the pattern in the first homology of the solid torus, equivalently its algebraic intersection number with a meridional disk.

The $(p,q)$-cable of $K$ uses the $(p,q)$ torus knot as a pattern in a tubular neighborhood of $K$.

For a pattern of winding number $w$,whenever the relevant signatures are defined by nonsingular forms. The usual averaged convention extends the identity through roots.

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

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

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.

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 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.

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 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.