8.1 The HOMFLY polynomial: definition and properties
2 min read•july 22, 2024
The is a powerful tool in knot theory, generalizing both the Alexander and Jones polynomials. It's defined using , which relate polynomials of links differing at a single crossing, allowing us to compute it for various knots and links.
Calculating the HOMFLY polynomial involves applying skein relations recursively until a link is reduced to a combination of unknots. This process helps distinguish between different knots and links, making it a valuable invariant in the study of knot theory.
Definition and Computation of the HOMFLY Polynomial
Definition of HOMFLY polynomial
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Denoted as ###p([l](https://www.fiveableKeyTerm:l))_0###, of (knots and links)
Generalizes and
Defined using skein relations relate polynomials of links differing at a single crossing
L+, L−, L0 represent links differing at a single crossing
l and [m](https://www.fiveableKeyTerm:m) are variables in the polynomial
P([U](https://www.fiveableKeyTerm:u))=1, where U is the (simplest knot)
Computation for simple knots
Apply skein relations recursively until link is reduced to combination of unknots
At each crossing, use skein relation to express polynomial in terms of simpler links
Continue process until link is fully simplified
(31) example:
Apply skein relation at one of the crossings
Express polynomial in terms of unknot and (two linked circles)
Use skein relation again on Hopf link to express it in terms of unknots
Simplify resulting expression to obtain HOMFLY polynomial for trefoil knot
Relationship to other polynomials
HOMFLY polynomial generalizes both Alexander and Jones polynomials
Setting l=i and m=i(t1/2−t−1/2) yields Jones polynomial
Setting l=i and m=i(t1/2−t−1/2), then substituting t=−s−2, yields Alexander polynomial multiplied by (−1)w(L)s−w(L), where w(L) is of link L (sum of crossing signs)
HOMFLY polynomial distinguishes more links than Alexander or Jones polynomials individually (figure-eight knot and its mirror image)
Invariance under Reidemeister moves
To prove invariance, show polynomial remains unchanged under each move
:
Use skein relation to express polynomial of link with twist in terms of polynomial of link without twist
Show resulting expression is equal to original polynomial
:
Apply skein relation twice to link with two overlapping strands
Show resulting expression is equal to polynomial of link without overlapping strands
:
Apply skein relation to each side of move
Show resulting expressions are equal, demonstrating invariance of polynomial under move