Arf invariant (knot)
Encyclopedia
In the mathematical field of knot theory
, the Arf invariant of a knot, named after Cahit Arf
, is a knot invariant
obtained from a quadratic form associated to a Seifert surface
. If F is a Seifert surface of a knot, then the homology group H1(F, Z/2Z) has a quadratic form whose value is the number of full twists mod 2 in a neighborhood of an imbedded circle representing an element of the homology group. The Arf invariant
of this quadratic form is the Arf invariant of the knot.
be a Seifert matrix of the knot, constructed from a set of curves on a Seifert surface
of genus g which represent a basis for the first homology
of the surface. This means that V is a 2g × 2g matrix with the property that V − VT is a symplectic matrix. The Arf invariant of the knot is the residue of
Specifically, if
, is a symplectic basis for the intersection form on the Seifert surface, then
where
denotes the positive pushoff of a.
.
We define two knots to be pass equivalent if they are related by a finite sequence of pass-moves, which are illustrated below: (no figure right now)
Every knot is pass-equivalent to either the unknot
or the trefoil
; these two knots are not pass-equivalent and additionally, the right- and left-handed trefoils are pass-equivalent.
Now we can define the Arf invariant of a knot to be 0 if it is pass-equivalent to the unknot, or 1 if it is pass-equivalent to the trefoil. This definition is equivalent to the one above.
showed that the Arf invariant can be obtained by taking the partition function of a signed planar graph associated to a knot diagram.
be the Alexander polynomial of the knot. Then the Arf invariant is the residue of
modulo 2, where r = 0 for n odd, and r = 1 for n even.
Kunio Murasugi proved that the Arf invariant is zero if and only if Δ(−1)
±1 modulo 8.
Knot theory
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...
, the Arf invariant of a knot, named after Cahit Arf
Cahit Arf
Cahit Arf was a Turkish mathematician. He is known for the Arf invariant of a quadratic form in characteristic 2 in topology, the Hasse–Arf theorem in ramification theory, Arf semigroups, and Arf rings.-Biography:Cahit Arf was born on 11 October 1910 in Selanik , which was then...
, is a knot invariant
Knot invariant
In the mathematical field of knot theory, a knot invariant is a quantity defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some invariants are indeed numbers, but invariants can range from the...
obtained from a quadratic form associated to a Seifert surface
Seifert surface
In mathematics, a Seifert surface is a surface whose boundary is a given knot or link.Such surfaces can be used to study the properties of the associated knot or link. For example, many knot invariants are most easily calculated using a Seifert surface...
. If F is a Seifert surface of a knot, then the homology group H1(F, Z/2Z) has a quadratic form whose value is the number of full twists mod 2 in a neighborhood of an imbedded circle representing an element of the homology group. The Arf invariant
Arf invariant
In mathematics, the Arf invariant of a nonsingular quadratic form over a field of characteristic 2 was defined by when he started the systematic study of quadratic forms over arbitrary fields of characteristic2. The Arf invariant is the substitute, in...
of this quadratic form is the Arf invariant of the knot.
Definition by Seifert matrix
Let be a Seifert matrix of the knot, constructed from a set of curves on a Seifert surfaceSeifert surface
In mathematics, a Seifert surface is a surface whose boundary is a given knot or link.Such surfaces can be used to study the properties of the associated knot or link. For example, many knot invariants are most easily calculated using a Seifert surface...
of genus g which represent a basis for the first homology
Homology (mathematics)
In mathematics , homology is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group...
of the surface. This means that V is a 2g × 2g matrix with the property that V − VT is a symplectic matrix. The Arf invariant of the knot is the residue of
Specifically, if
, is a symplectic basis for the intersection form on the Seifert surface, thenwhere
denotes the positive pushoff of a.Definition by pass equivalence
This approach to the Arf invariant is due to Louis KauffmanLouis Kauffman
Louis H. Kauffman is an American mathematician, topologist, and professor of Mathematics in the Department of Mathematics, Statistics, and Computer science at the University of Illinois at Chicago...
.
We define two knots to be pass equivalent if they are related by a finite sequence of pass-moves, which are illustrated below: (no figure right now)
Every knot is pass-equivalent to either the unknot
Unknot
The unknot arises in the mathematical theory of knots. Intuitively, the unknot is a closed loop of rope without a knot in it. A knot theorist would describe the unknot as an image of any embedding that can be deformed, i.e. ambient-isotoped, to the standard unknot, i.e. the embedding of the...
or the trefoil
Trefoil
Trefoil is a graphic form composed of the outline of three overlapping rings used in architecture and Christian symbolism...
; these two knots are not pass-equivalent and additionally, the right- and left-handed trefoils are pass-equivalent.
Now we can define the Arf invariant of a knot to be 0 if it is pass-equivalent to the unknot, or 1 if it is pass-equivalent to the trefoil. This definition is equivalent to the one above.
Definition by partition function
Vaughan JonesVaughan Jones
Sir Vaughan Frederick Randal Jones, KNZM, FRS, FRSNZ is a New Zealand mathematician, known for his work on von Neumann algebras, knot polynomials and conformal field theory. He was awarded a Fields Medal in 1990, and famously wore a New Zealand rugby jersey when he accepted the prize...
showed that the Arf invariant can be obtained by taking the partition function of a signed planar graph associated to a knot diagram.
Definition by Alexander polynomial
This approach to the Arf invariant is by Raymond Robertello. Let
be the Alexander polynomial of the knot. Then the Arf invariant is the residue of
modulo 2, where r = 0 for n odd, and r = 1 for n even.
Kunio Murasugi proved that the Arf invariant is zero if and only if Δ(−1)
±1 modulo 8.