L11a525

(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_22,19,13,20 X_16,9,17,10 X_10,22,11,21 X_20,12,21,11 X_18,5,19,6 X₂₇₃₈ X_4,13,5,14 X_6,17,1,18
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 5, -6, 7, -3}, {10, -2, 3, -5, 11, -8, 4, -7, 6, -4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- u 2 v 3 + 2 u v 3 - u w v 3 + w v 3 - v 3 + 3 u 2 v 2 -2 u w 2 v 2 + 2 w 2 v 2 -5 u v 2 -2 u 2 w v 2 + 6 u w v 2 -3 w v 2 + v 2 -2 u 2 v - u 2 w 2 v + 5 u w 2 v -3 w 2 v + 2 u v + 3 u 2 w v -6 u w v + 2 w v + u 2 w 2 -2 u w 2 + w 2 - u 2 w + u w$ (db)
Jones polynomial	$q -3 + 7 q -1 -12 q -2 + 17 q -3 -18 q -4 + 20 q -5 -16 q -6 + 13 q -7 -8 q -8 + 4 q -9 - q -10$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$- a 10 + 4 z 2 a 8 + a 8 z -2 + 3 a 8 -3 z 4 a 6 -4 z 2 a 6 -2 a 6 z -2 -5 a 6 -3 z 4 a 4 - z 2 a 4 + a 4 z -2 + 2 a 4 - z 4 a 2 + 2 z 2 a 2 + a 2 + z 2$ (db)
Kauffman polynomial	$z 7 a 11 -3 z 5 a 11 + 3 z 3 a 11 - z a 11 + 4 z 8 a 10 -14 z 6 a 10 + 15 z 4 a 10 -6 z 2 a 10 + 2 a 10 + 5 z 9 a 9 -13 z 7 a 9 + 2 z 5 a 9 + 10 z 3 a 9 -3 z a 9 + 2 z 10 a 8 + 8 z 8 a 8 -46 z 6 a 8 + 59 z 4 a 8 -34 z 2 a 8 - a 8 z -2 + 10 a 8 + 12 z 9 a 7 -30 z 7 a 7 + 12 z 5 a 7 + 8 z 3 a 7 -8 z a 7 + 2 a 7 z -1 + 2 z 10 a 6 + 14 z 8 a 6 -56 z 6 a 6 + 66 z 4 a 6 -41 z 2 a 6 -2 a 6 z -2 + 12 a 6 + 7 z 9 a 5 -7 z 7 a 5 -9 z 5 a 5 + 12 z 3 a 5 -6 z a 5 + 2 a 5 z -1 + 10 z 8 a 4 -18 z 6 a 4 + 15 z 4 a 4 -8 z 2 a 4 - a 4 z -2 + 4 a 4 + 9 z 7 a 3 -13 z 5 a 3 + 9 z 3 a 3 + 6 z 6 a 2 -6 z 4 a 2 + 4 z 2 a 2 - a 2 + 3 z 5 a -2 z 3 a + z 4 - z 2$ (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 L11a525. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -3$	$i = -1$
$r = -9$	${\mathbb Z}$
$r = -8$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -7$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -6$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{6}$
$r = -5$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = -4$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{10}$
$r = -3$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -2$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{9}$
$r = -1$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 0$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{5}$
$r = 1$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 2$	${\mathbb Z}_2$	${\mathbb Z}$