L11a8

(Knotscape image)

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

Visit L11a8's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_20,9,21,10 X₈₄₉₃ X_18,22,19,21 X_14,12,15,11 X_12,5,13,6 X_22,13,5,14 X_10,19,11,20 X_2,16,3,15
Gauss code	{1, -11, 5, -3}, {8, -1, 2, -5, 4, -10, 7, -8, 9, -7, 11, -2, 3, -6, 10, -4, 6, -9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + u 5 + 6 v u 4 -6 u 4 -14 v u 3 + 14 u 3 + 14 v u 2 -14 u 2 -6 v u + 6 u + v -1$ (db)
Jones polynomial	$-q^{7/2}+5 q^{5/2}-11 q^{3/2}+17 \sqrt{q}-\frac{25}{\sqrt{q}}+\frac{27}{q^{3/2}}-\frac{27}{q^{5/2}}+\frac{23}{q^{7/2}}-\frac{17}{q^{9/2}}+\frac{10}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z a 7 + 3 z 3 a 5 + 2 z a 5 - a 5 z -1 -3 z 5 a 3 -5 z 3 a 3 + 3 a 3 z -1 + z 7 a + 3 z 5 a + 4 z 3 a - z a -2 a z -1 - z 5 a -1 - z 3 a -1$ (db)
Kauffman polynomial	$-2 a 4 z 10 -2 a 2 z 10 -7 a 5 z 9 -15 a 3 z 9 -8 a z 9 -8 a 6 z 8 -22 a 4 z 8 -27 a 2 z 8 -13 z 8 -4 a 7 z 7 + 3 a 5 z 7 + 8 a 3 z 7 -10 a z 7 -11 z 7 a -1 - a 8 z 6 + 17 a 6 z 6 + 59 a 4 z 6 + 61 a 2 z 6 -5 z 6 a -2 + 15 z 6 + 8 a 7 z 5 + 18 a 5 z 5 + 35 a 3 z 5 + 41 a z 5 + 15 z 5 a -1 - z 5 a -3 + 2 a 8 z 4 -14 a 6 z 4 -47 a 4 z 4 -39 a 2 z 4 + 4 z 4 a -2 -4 z 4 -6 a 7 z 3 -18 a 5 z 3 -30 a 3 z 3 -25 a z 3 -7 z 3 a -1 - a 8 z 2 + 6 a 6 z 2 + 16 a 4 z 2 + 11 a 2 z 2 + 2 z 2 + 2 a 7 z + 4 a 5 z + a 3 z - a z - a 6 -3 a 4 -3 a 2 + a 5 z -1 + 3 a 3 z -1 + 2 a z -1$ (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 L11a8. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -7$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -4$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = -3$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -2$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{13}$
$r = -1$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{14}$	${\mathbb Z}^{14}$
$r = 0$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{15}$
$r = 1$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 2$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 4$	${\mathbb Z}_2$	${\mathbb Z}$