L11a526

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_14,3,15,4 X_12,15,7,16 X_10,22,11,21 X_16,9,17,10 X_20,12,21,11 X₂₇₃₈ X_22,17,13,18 X_6,13,1,14 X_4,20,5,19 X_18,6,19,5
Gauss code	{1, -7, 2, -10, 11, -9}, {7, -1, 5, -4, 6, -3}, {9, -2, 3, -5, 8, -11, 10, -6, 4, -8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 3 + v w 2 u 3 - w 2 u 3 + 2 v u 3 + v 2 w u 3 -3 v w u 3 + 2 w u 3 - u 3 + 2 v 2 u 2 -3 v w 2 u 2 + 2 w 2 u 2 -4 v u 2 -3 v 2 w u 2 + 9 v w u 2 -4 w u 2 + u 2 -2 v 2 u - v 2 w 2 u + 4 v w 2 u -2 w 2 u + 3 v u + 4 v 2 w u -9 v w u + 3 w u + v 2 + v 2 w 2 -2 v w 2 + w 2 - v -2 v 2 w + 3 v w - w$ (db)
Jones polynomial	$q 3 -4 q 2 + 10 q -16 + 23 q -1 -25 q -2 + 27 q -3 -22 q -4 + 17 q -5 -10 q -6 + 4 q -7 - q -8$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$- z 4 a 6 - z 2 a 6 -2 a 6 + z 6 a 4 + 2 z 4 a 4 + 6 z 2 a 4 + a 4 z -2 + 6 a 4 + z 6 a 2 - z 4 a 2 -6 z 2 a 2 -2 a 2 z -2 -7 a 2 -2 z 4 + z -2 + 3 + z 2 a -2$ (db)
Kauffman polynomial	$3 a 4 z 10 + 3 a 2 z 10 + 10 a 5 z 9 + 18 a 3 z 9 + 8 a z 9 + 13 a 6 z 8 + 20 a 4 z 8 + 15 a 2 z 8 + 8 z 8 + 9 a 7 z 7 -10 a 5 z 7 -35 a 3 z 7 -12 a z 7 + 4 z 7 a -1 + 4 a 8 z 6 -25 a 6 z 6 -64 a 4 z 6 -54 a 2 z 6 + z 6 a -2 -18 z 6 + a 9 z 5 -13 a 7 z 5 -5 a 5 z 5 + 18 a 3 z 5 + a z 5 -8 z 5 a -1 -4 a 8 z 4 + 24 a 6 z 4 + 73 a 4 z 4 + 62 a 2 z 4 -2 z 4 a -2 + 15 z 4 - a 9 z 3 + 7 a 7 z 3 + 13 a 5 z 3 + 3 a 3 z 3 + 2 a z 3 + 4 z 3 a -1 -13 a 6 z 2 -42 a 4 z 2 -40 a 2 z 2 + z 2 a -2 -10 z 2 -2 a 7 z -6 a 5 z -7 a 3 z -3 a z + 4 a 6 + 13 a 4 + 13 a 2 + 5 + 2 a 3 z -1 + 2 a z -1 - a 4 z -2 -2 a 2 z -2 - z -2$ (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 =

-2 is the signature of L11a526. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -3$	$i = -1$
$r = -7$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -4$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = -3$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -2$	${\mathbb Z}^{15}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{14}$
$r = -1$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{13}$
$r = 0$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{14}$
$r = 1$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 2$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 4$	${\mathbb Z}_2$	${\mathbb Z}$