8.1 The HOMFLY polynomial: definition and properties

The HOMFLY polynomial is a powerful tool in knot theory, generalizing both the Alexander and Jones polynomials. It's defined using skein relations, 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

• Denoted as $P(L)$, two-variable polynomial invariant of oriented links (knots and links)
• Generalizes Alexander polynomial and Jones polynomial
• Defined using skein relations relate polynomials of links differing at a single crossing
• Skein relations for HOMFLY polynomial:
  • $lP(L_+) + l^{-1}P(L_-) + mP(L_0) = 0$
    • $L_+$, $L_-$, $L_0$ represent links differing at a single crossing
    • $l$ and $m$ are variables in the polynomial
  • $P(U) = 1$, where $U$ is the unknot (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
• Trefoil knot $(3_1)$ example:
  1. Apply skein relation at one of the crossings
  2. Express polynomial in terms of unknot and Hopf link (two linked circles)
  3. Use skein relation again on Hopf link to express it in terms of unknots
  4. 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 $m = i(t^{1/2} - t^{-1/2})$ yields Jones polynomial
  • Setting $l = i$ and $m = i(t^{1/2} - t^{-1/2})$, then substituting $t = -s^{-2}$, yields Alexander polynomial multiplied by $(-1)^{w(L)}s^{-w(L)}$, where $w(L)$ is writhe 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
• Reidemeister move I:
  • 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
• Reidemeister move II:
  • Apply skein relation twice to link with two overlapping strands
  • Show resulting expression is equal to polynomial of link without overlapping strands
• Reidemeister move III:
  • Apply skein relation to each side of move
  • Show resulting expressions are equal, demonstrating invariance of polynomial under move