Colored Jones polynomial

The colored Jones polynomial is an invariant from the mathematical field of knot theory . It depends on a parameter and assigns an intertwining a Laurent polynomial in one variable to. For we get the Jones polynomial . ${\displaystyle N\in \mathbb {N} }$ ${\displaystyle L}$ ${\displaystyle J_{N}(L,q)}$ ${\displaystyle q}$ ${\displaystyle N=2}$

definition

The colored Jones polynomial is the quantum invariant corresponding to the N-dimensional irreducible representation of . It becomes explicit with the R matrix ${\displaystyle sl(2,\mathbb {C} )}$

${\displaystyle R_{kl}^{ij}:=\sum _{m=0}^{N-1-j,i}\delta _{l,i+m}\delta _{k,j-m}{\frac {\left\{l\right\}!\left\{N-1-k\right\}!}{\left\{i\right\}!\left\{m\right\}!\left\{N-1-j\right\}!}}q^{(-i-{\frac {N-1}{2}})(j-{\frac {N-1}{2}})-m{\frac {i-j}{2}}-{\frac {m(m+1)}{4}}}}$

and constructed by the given isomorphism , see quantum invariant # construction via R-matrices . Here is and . ${\displaystyle \mu (e_{j})=\sum _{i=0}^{N-1}\delta _{ij}q^{\frac {2i-N+1}{2}}e_{i}}$ ${\displaystyle \mu \colon \mathbb {C} ^{N}\to \mathbb {C} ^{N}}$ ${\displaystyle \left\{m\right\}:=q^{\frac {m}{2}}-q^{-{\frac {m}{2}}}}$ ${\displaystyle \left\{m\right\}!=\left\{1\right\}\left\{2\right\}\ldots \left\{m\right\}}$

Alternatively, you can of than the Jones polynomial to define existing parallel entanglement entanglement. However, this approach is completely impractical for concrete calculations because the number of crossovers increases quadratically in . ${\displaystyle J_{N}(L,q)}$ ${\displaystyle N-1}$ ${\displaystyle L}$ ${\displaystyle N}$

example

The colored Jones polynomial of the shamrock is

${\displaystyle J_{N}(L,q)={\frac {q^{\frac {1-n}{2}}}{1-q^{-1}}}\sum _{k=0}^{n-1}q^{-kn}(1-q^{-n})(1-q^{1-n})\ldots (1-q^{k-n})}$.

The colored Jones polynomial of the figure eight is

${\displaystyle J_{N}(L,q)=q^{1-N}\sum _{n=0}^{N-1}\sum _{k=0}^{n}\left(\prod _{i=0}^{k-1}{\frac {1-q^{n-i}}{1-q^{i+1}}}\right)q^{n+k(k+1)}\left[\prod _{j=1}^{n}(1-q^{j-N})\right]\left[\prod _{i=1}^{n-k}(1-q^{k+i-N})\right]}$.

properties

• The colored Jones polynomial is multiplicative with affiliated sum : . ${\displaystyle J_{N}(L_{1}\sharp L_{2},q)=J_{N}(L_{1},q)J_{N}(L_{2},q)}$
• The colored Jones polynomial satisfies a recurrence relation.

Kashaev invariant

The Kashaev invariant is the value of the colored Jones polynomial at the Nth root of unit:

${\displaystyle \langle L\rangle _{N}:=J_{N}(L,e^{\frac {2\pi i}{N}})}$.

The volume conjecture establishes a connection between the Kashaev invariant and the complex volume of a hyperbolic knot .

literature

• Wladimir Turajew : Quantum invariants of knots and 3-manifolds. Second revised edition. de Gruyter Studies in Mathematics, 18. Walter de Gruyter & Co., Berlin, 2010. ISBN 978-3-11-022183-1
• PM Melvin, HR Morton: The Colored Jones function. Comm. Math. Phys. 169 (1995) no. 3: 501-520.

Web links

• The colored Jones polynomial

Individual evidence

1. ↑ Garoufalidis, Stavros; Lê, Thang TQ The colored Jones function is q-holonomic. Geom. Topol. 9: 1253-1293 (2005)