Knot theory studies embeddings of circles in three-dimensional manifolds and their higher-dimensional analogues.
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 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 , the torus knot winds times meridionally and times longitudinally on an unknotted torus in .
A satellite knot is obtained by embedding a patterned solid torus into a tubular neighborhood of a companion knot .
A pattern is a knot inside a solid torus. Embedding that torus as a tubular neighborhood of produces the satellite .
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.
For a pattern of winding number ,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 is a compact connected oriented surface with oriented boundary .
For oriented curves on a Seifert surface, the Seifert form iswhere is a positive normal push-off of .
A Seifert matrix represents the Seifert form in an integral basis of . Its skew-symmetrization represents the intersection form of the surface and is unimodular.
The knot determinant is . It is also the order of the first homology of the two-fold cover of branched over .
For , 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 -dimensional space is metabolic when it vanishes on a -dimensional subspace called a metabolizer.
The algebraic concordance group over a field is the Witt group of nonsingular Seifert forms over , modulo metabolic forms.
An isometric structure consists of a finite-dimensional vector space , a nonsingular symmetric bilinear form , and a -isometry with the required nondegeneracy at . Metabolic isometric structures are quotiented out to form .
The Witt group of isometric structures identifies two isometric structures when their orthogonal difference is metabolic.
For an irreducible symmetric Laurent polynomial , the primary component isfor sufficiently large . Distinct symmetric primary components are orthogonal, so restriction defines a projection .
Extending an algebraic-concordance class from to a p-adic field detects torsion invisible over the real numbers. For , an odd-dimensional second-residue form generates the order-four part of the local Witt group.
Two knots are concordant when they cobound a smoothly embedded annulus in . Connected sum makes concordance classes into an abelian group.
A ribbon disk is a slice disk whose radial Morse function has no interior local maxima. It can be built from disks by attaching bands.
The slice genus is the smallest genus of a smooth compact connected oriented surface properly embedded in with boundary .
A knot is doubly slice when it is the equatorial cross-section of an unknotted smooth two-sphere in .
The double slice genus is the smallest genus of an unknotted closed orientable surface in whose transverse equatorial cross-section is .
The two-fold branched cover of a knot is the double cover of branched along the knot. Its first homology has a nonsingular linking form and, when finite, has order .
For a rational homology three-sphere , the linking form is the nonsingular pairingReversing the orientation of negates this form.
An amphichiral knot is ambient-isotopic to its mirror. Positive and negative amphichirality distinguish whether the symmetry preserves or reverses the knot orientation.
The Arf invariant is a -valued concordance invariant obtained from the quadratic refinement of the mod-two intersection form on a Seifert surface.
An -twist spin removes a trivial arc from a knot and spins the resulting knotted arc around the boundary of a three-ball in , inserting full twists during one revolution.
Zeeman's theorem says that the complement of an -twist spin fibers over the circle with fiber the punctured -fold cyclic branched cover of the original knot. In particular, the - and -twist spins are unknotted.
A surface knot is a smooth embedding of a closed connected surface in a four-manifold. In the narrow standard usage, a 2-knot is an embedded two-sphere in .
A banded-unlink diagram consists of an unlink and disjoint bands whose simultaneous surgery produces another unlink. Capping the lower and upper unlinks by disks and tracing the band surgeries constructs a surface in .
The Seifert longitude is the zero-linking parallel of a knot on the boundary of its tubular neighborhood. It is the framing induced by a Seifert surface.
A link is a smooth embedding of a finite disjoint union of circles in , considered up to ambient isotopy. A one-component link is a knot.
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Knot theory is a branch of mathematics that studies mathematical knots, which are loops in three-dimensional space that do not intersect themselves. It is a part of the field of topology, specifically dealing with the properties of these loops that remain unchanged through continuous deformations, such as stretching, twisting, and bending, but not cutting or gluing. In knot theory, a "knot" is defined as an embedded circle in three-dimensional Euclidean space \( \mathbb{R}^3 \).