Many knot polynomials are computed using skein relations , which allow one to change the different crossings of a knot to get simpler knots. In the mathematical field of knot theory , a knot polynomial is a knot invariant in the form of a polynomial whose coefficients encode some of the properties of a given knot .
History
The first knot polynomial, the Alexander polynomial , was introduced by James Waddell Alexander II in 1923. Other knot polynomials were not found until almost 60 years later.
In the 1960s, John Conway came up with a skein relation for a version of the Alexander polynomial, usually referred to as the Alexander–Conway polynomial . The significance of this skein relation was not realized until the early 1980s, when Vaughan Jones discovered the Jones polynomial . This led to the discovery of more knot polynomials, such as the so-called HOMFLY polynomial .
Soon after Jones' discovery, Louis Kauffman noticed the Jones polynomial could be computed by means of a partition function (state-sum model), which involved the bracket polynomial , an invariant of framed knots . This opened up avenues of research linking knot theory and statistical mechanics .
In the late 1980s, two related breakthroughs were made. Edward Witten demonstrated that the Jones polynomial, and similar Jones-type invariants, had an interpretation in Chern–Simons theory . Viktor Vasilyev and Mikhail Goussarov started the theory of finite type invariants of knots. The coefficients of the previously named polynomials are known to be of finite type (after perhaps a suitable "change of variables").
In recent years, the Alexander polynomial has been shown to be related to Floer homology . The graded Euler characteristic of the knot Floer homology of Peter Ozsváth and Zoltan Szabó is the Alexander polynomial.
Examples
Alexander–Briggs notation Alexander polynomial
Δ
(
t
)
{\displaystyle \Delta (t)}
Conway polynomial
∇
(
z
)
{\displaystyle \nabla (z)}
Jones polynomial
V
(
q
)
{\displaystyle V(q)}
HOMFLY polynomial
H
(
a
,
z
)
{\displaystyle H(a,z)}
0
1
{\displaystyle 0_{1}}
Unknot )
1
{\displaystyle 1}
1
{\displaystyle 1}
1
{\displaystyle 1}
1
{\displaystyle 1}
3
1
{\displaystyle 3_{1}}
Trefoil Knot )
t
−
1
+
t
−
1
{\displaystyle t-1+t^{-1}}
z
2
+
1
{\displaystyle z^{2}+1}
q
−
1
+
q
−
3
−
q
−
4
{\displaystyle q^{-1}+q^{-3}-q^{-4}}
−
a
4
+
a
2
z
2
+
2
a
2
{\displaystyle -a^{4}+a^{2}z^{2}+2a^{2}}
4
1
{\displaystyle 4_{1}}
Figure-eight Knot )
−
t
+
3
−
t
−
1
{\displaystyle -t+3-t^{-1}}
−
z
2
+
1
{\displaystyle -z^{2}+1}
q
2
−
q
+
1
−
q
−
1
+
q
−
2
{\displaystyle q^{2}-q+1-q^{-1}+q^{-2}}
a
2
+
a
−
2
−
z
2
−
1
{\displaystyle a^{2}+a^{-2}-z^{2}-1}
5
1
{\displaystyle 5_{1}}
Cinquefoil Knot )
t
2
−
t
+
1
−
t
−
1
+
t
−
2
{\displaystyle t^{2}-t+1-t^{-1}+t^{-2}}
z
4
+
3
z
2
+
1
{\displaystyle z^{4}+3z^{2}+1}
q
−
2
+
q
−
4
−
q
−
5
+
q
−
6
−
q
−
7
{\displaystyle q^{-2}+q^{-4}-q^{-5}+q^{-6}-q^{-7}}
−
a
6
z
2
−
2
a
6
+
a
4
z
4
+
4
a
4
z
2
+
3
a
4
{\displaystyle -a^{6}z^{2}-2a^{6}+a^{4}z^{4}+4a^{4}z^{2}+3a^{4}}
3
1
#
3
1
{\displaystyle 3_{1}\#3_{1}}
Granny Knot )
(
t
−
1
+
t
−
1
)
2
{\displaystyle \left(t-1+t^{-1}\right)^{2}}
(
z
2
+
1
)
2
{\displaystyle \left(z^{2}+1\right)^{2}}
(
q
−
1
+
q
−
3
−
q
−
4
)
2
{\displaystyle \left(q^{-1}+q^{-3}-q^{-4}\right)^{2}}
(
−
a
4
+
a
2
z
2
+
2
a
2
)
2
{\displaystyle \left(-a^{4}+a^{2}z^{2}+2a^{2}\right)^{2}}
3
1
#
3
1
∗
{\displaystyle 3_{1}\#3_{1}^{*}}
Square Knot )
(
t
−
1
+
t
−
1
)
2
{\displaystyle \left(t-1+t^{-1}\right)^{2}}
(
z
2
+
1
)
2
{\displaystyle \left(z^{2}+1\right)^{2}}
(
q
−
1
+
q
−
3
−
q
−
4
)
(
q
+
q
3
−
q
4
)
{\displaystyle \left(q^{-1}+q^{-3}-q^{-4}\right)\left(q+q^{3}-q^{4}\right)}
(
−
a
4
+
a
2
z
2
+
2
a
2
)
×
{\displaystyle \left(-a^{4}+a^{2}z^{2}+2a^{2}\right)\times }
(
−
a
−
4
+
a
−
2
z
−
2
+
2
a
−
2
)
{\displaystyle \left(-a^{-4}+a^{-2}z^{-2}+2a^{-2}\right)}
Alexander–Briggs notation organizes knots by their crossing number.
Alexander polynomials and Conway polynomials can not recognize the difference of left-trefoil knot and right-trefoil knot.
The left-trefoil knot.
The right-trefoil knot.
So we have the same situation as the granny knot and square knot since the addition of knots in
R
3
{\displaystyle \mathbb {R} ^{3}}
knot polynomials .
See also
Specific knot polynomials
Graph polynomial , a similar class of polynomial invariants in graph theoryTutte polynomial , a special type of graph polynomial related to the Jones polynomialSkein relation for a formal definition of the Alexander polynomial, with a worked-out example.
Further reading
.svg)
