HOMFLY polynomial

The HOMFLY polynomial , also HOMFLY-PT polynomial , is a generalization of the Alexander polynomial and Jones polynomial in node theory , which assigns a polynomial in the variables and to each node . It is also an example of a quantum invariant . ${\displaystyle m}$ ${\displaystyle l}$

The name is made up of the initials of the co-discoverers: Jim Hoste, Adrian Ocneanu, Kenneth Millett, Peter Freyd , WBR Lickorish , David N. Yetter, Józef H. Przytycki, Paweł Traczyk.

definition

The polynomial is defined as follows:

${\displaystyle P(\mathrm {unknot} )=1,\,}$

${\displaystyle \ell P(L_{+})+\ell ^{-1}P(L_{-})+mP(L_{0})=0,\,}$

where are links formed by crossing and smoothing. ${\displaystyle L_{+},L_{-},L_{0}}$

The HOMFLY polynomial of a link that is the disjoint union of two links and is ${\displaystyle L}$ ${\displaystyle L_{1}}$ ${\displaystyle L_{2}}$

${\displaystyle P(L)={\frac {-(\ell +\ell ^{-1})}{m}}P(L_{1})P(L_{2}).}$

properties

It applies

${\displaystyle P(L_{1}\#L_{2})=P(L_{1})P(L_{2}),\,}$,

where denotes the connected sum ; therefore the HOMFLY polynomial of a compound node is the product of the HOMFLY polynomials of its components. ${\displaystyle \#}$

Also is

${\displaystyle P_{K}(\ell ,m)=P_{{\text{Mirror Image}}(K)}(\ell ^{-1},m),\,}$,

so the HOMFLY polynomial can often be used to distinguish between two nodes of different chirality, even though there are chiral pairs of nodes that have the same HOMFLY polynomial, e.g. B. 9 ₄₂ and 10 ₇₁ .

The Jones polynomial and the Alexander polynomial can be calculated from the HOMFLY polynomial as follows: ${\displaystyle V(t)}$ ${\displaystyle \Delta (t)\,}$

${\displaystyle V(t)=P(l=t^{-1},m=t^{-1/2}-t^{1/2}),\,}$

${\displaystyle \Delta (t)=P(l=1,m=t^{-1/2}-t^{1/2}).\,}$

More generally, can the - quantum invariant from the HOMFLY polynomial calculated. ${\displaystyle {{\mathfrak {s}}l}_{N}}$

literature

• Louis H. Kauffman : Formal knot theory, 1983

Web links

• Gukov, Saberi: Lectures on knot homology and quantum curves

Individual evidence

1. ↑ Freyd, P .; Yetter, D., Hoste, J., Lickorish, WBR, Millett, K., and Ocneanu, A. (1985). "A New Polynomial Invariant of Knots and Links". Bulletin of the American Mathematical Society 12 (2): 239-246
2. ↑ P. Ramadevi, TR Govindarajan, RK Kaul: Chirality of Knots 9 ₄₂ and 10 ₇₁ and Chern-Simons Theory.