L11n88

(Knotscape image)

See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n88's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_3,13,4,12 X_7,16,8,17 X_17,22,18,5 X_9,15,10,14 X_19,10,20,11 X_21,9,22,8 X_13,18,14,19 X_15,21,16,20 X₂₅₃₆ X_11,1,12,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 7, -5, 6, -11, 2, -8, 5, -9, 3, -4, 8, -6, 9, -7, 4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + u 5 + 4 v u 4 -4 u 4 -7 v u 3 + 7 u 3 + 7 v u 2 -7 u 2 -4 v u + 4 u + v -1$ (db)
Jones polynomial	$q^{9/2}-4 q^{7/2}+8 q^{5/2}-13 q^{3/2}+15 \sqrt{q}-\frac{17}{\sqrt{q}}+\frac{15}{q^{3/2}}-\frac{12}{q^{5/2}}+\frac{8}{q^{7/2}}-\frac{3}{q^{9/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z 7 - a 3 z 5 + 4 a z 5 -2 z 5 a -1 -3 a 3 z 3 + 7 a z 3 -5 z 3 a -1 + z 3 a -3 + a 5 z -5 a 3 z + 7 a z -4 z a -1 + z a -3 + a 5 z -1 -3 a 3 z -1 + 4 a z -1 -2 a -1 z -1$ (db)
Kauffman polynomial	$-3 a z 9 -3 z 9 a -1 -10 a 2 z 8 -6 z 8 a -2 -16 z 8 -10 a 3 z 7 -13 a z 7 -7 z 7 a -1 -4 z 7 a -3 -3 a 4 z 6 + 22 a 2 z 6 + 13 z 6 a -2 - z 6 a -4 + 39 z 6 + 21 a 3 z 5 + 48 a z 5 + 37 z 5 a -1 + 10 z 5 a -3 -6 a 4 z 4 -27 a 2 z 4 -5 z 4 a -2 + 2 z 4 a -4 -28 z 4 -6 a 5 z 3 -28 a 3 z 3 -49 a z 3 -35 z 3 a -1 -8 z 3 a -3 + 7 a 4 z 2 + 17 a 2 z 2 - z 2 a -4 + 11 z 2 + 4 a 5 z + 16 a 3 z + 22 a z + 13 z a -1 + 3 z a -3 - a 4 -3 a 2 - a -2 -2- a 5 z -1 -3 a 3 z -1 -4 a z -1 -2 a -1 z -1$ (db)

The coefficients of the monomials

t r q j

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 -2 r = s + 1

j -2 r = s -1

, where

s =

-1 is the signature of L11n88. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -4$	${\mathbb Z}^{3}$
$r = -3$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -1$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 0$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{10}$
$r = 1$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 2$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 4$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 5$	${\mathbb Z}_2$	${\mathbb Z}$