L11a527

(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_10,21,11,22 X_16,9,17,10 X_22,11,13,12 X_18,6,19,5 X_20,18,21,17 X₂₇₃₈ X_4,13,5,14 X_6,20,1,19
Gauss code	{1, -9, 2, -10, 7, -11}, {9, -1, 5, -4, 6, -3}, {10, -2, 3, -5, 8, -7, 11, -8, 4, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v w 2 u 3 + w 2 u 3 - v 2 w u 3 + 2 v w u 3 - w u 3 -2 v 2 u 2 - v 2 w 2 u 2 + 3 v w 2 u 2 -3 w 2 u 2 + 3 v u 2 + 4 v 2 w u 2 -7 v w u 2 + 4 w u 2 - u 2 + 3 v 2 u + v 2 w 2 u -3 v w 2 u + 2 w 2 u -3 v u -4 v 2 w u + 7 v w u -4 w u + u - v 2 + v + v 2 w -2 v w + w$ (db)
Jones polynomial	$- q 2 + 4 q -9 + 15 q -1 -19 q -2 + 23 q -3 -21 q -4 + 19 q -5 -13 q -6 + 8 q -7 -3 q -8 + q -9$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$z 2 a 8 + a 8 z -2 + 2 a 8 -3 z 4 a 6 -8 z 2 a 6 -2 a 6 z -2 -8 a 6 + 2 z 6 a 4 + 7 z 4 a 4 + 11 z 2 a 4 + a 4 z -2 + 7 a 4 + z 6 a 2 + z 4 a 2 - z 2 a 2 - a 2 - z 4 - z 2$ (db)
Kauffman polynomial	$z 6 a 10 -3 z 4 a 10 + 3 z 2 a 10 - a 10 + 3 z 7 a 9 -7 z 5 a 9 + 4 z 3 a 9 + 5 z 8 a 8 -10 z 6 a 8 + 6 z 4 a 8 -5 z 2 a 8 - a 8 z -2 + 4 a 8 + 5 z 9 a 7 -6 z 7 a 7 -3 z 5 a 7 + 6 z 3 a 7 -6 z a 7 + 2 a 7 z -1 + 2 z 10 a 6 + 10 z 8 a 6 -35 z 6 a 6 + 43 z 4 a 6 -31 z 2 a 6 -2 a 6 z -2 + 12 a 6 + 12 z 9 a 5 -21 z 7 a 5 + 8 z 5 a 5 + 7 z 3 a 5 -8 z a 5 + 2 a 5 z -1 + 2 z 10 a 4 + 15 z 8 a 4 -45 z 6 a 4 + 52 z 4 a 4 -31 z 2 a 4 - a 4 z -2 + 10 a 4 + 7 z 9 a 3 -4 z 7 a 3 -9 z 5 a 3 + 11 z 3 a 3 -3 z a 3 + 10 z 8 a 2 -17 z 6 a 2 + 13 z 4 a 2 -7 z 2 a 2 + 2 a 2 + 8 z 7 a -12 z 5 a + 5 z 3 a - z a + 4 z 6 -5 z 4 + z 2 + z 5 a -1 - z 3 a -1$ (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 =

-2 is the signature of L11a527. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -3$	$i = -1$
$r = -8$	${\mathbb Z}$
$r = -7$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{3}$
$r = -5$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -4$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{9}$
$r = -3$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -2$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{12}$
$r = -1$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 0$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{9}$
$r = 1$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 2$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 3$	${\mathbb Z}_2$	${\mathbb Z}$