# L8n3

## Contents

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8n3's page at Knotilus. Visit L8n3's page at the original Knot Atlas. L8n3 is $8^3_{7}$ in the Rolfsen table of links.

 Planar diagram presentation X6172 X5,12,6,13 X3849 X13,2,14,3 X14,7,15,8 X9,16,10,11 X11,10,12,5 X4,15,1,16 Gauss code {1, 4, -3, -8}, {-2, -1, 5, 3, -6, 7}, {-7, 2, -4, -5, 8, 6}

### Polynomial invariants

 Multivariable Alexander Polynomial (in u, v, w, ...) $\frac{u v^2 w^2-1}{\sqrt{u} v w}$ (db) Jones polynomial q−3 + q−5 + q−7 + q−9 (db) Signature -6 (db) HOMFLY-PT polynomial a10z−2 + a10−z4a8−5z2a8−2a8z−2−6a8 + z6a6 + 6z4a6 + 10z2a6 + a6z−2 + 5a6 (db) Kauffman polynomial a12 + a10z2 + a10z−2−3a10 + a9z5−5a9z3 + 6a9z−2a9z−1 + a8z6−6a8z4 + 11a8z2 + 2a8z−2−8a8 + a7z5−5a7z3 + 6a7z−2a7z−1 + a6z6−6a6z4 + 10a6z2 + a6z−2−5a6 (db)

### Khovanov Homology

 The coefficients of the monomials trqj are shown, along with their alternating sums χ (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j−2r = s + 1 or j−2r = s−1, where s = -6 is the signature of L8n3. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. Data:L8n3/KhovanovTable
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = −7 i = −5 i = −3 r = −6 ${\mathbb Z}$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ r = −5 ${\mathbb Z}$ r = −4 ${\mathbb Z}_2$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ r = −3 ${\mathbb Z}$ r = −2 ${\mathbb Z}_2$ ${\mathbb Z}$ r = −1 r = 0 ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory. See A Sample KnotTheory Session.

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).