L11a536

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X₂₅₃₆ X_18,12,19,11 X_10,3,11,4 X_4,9,1,10 X_16,7,17,8 X_8,15,5,16 X_22,18,15,17 X_12,22,13,21 X_20,14,21,13 X_14,20,9,19
Gauss code	{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -9, 10, -11}, {7, -6, 8, -3, 11, -10, 9, -8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 2 + 2 v w u 2 -2 w u 2 + 2 v x u 2 - v w x u 2 + w x u 2 - x u 2 + u 2 -2 v 2 u + 3 v u + 2 v 2 w u -4 v w u + 2 w u + 2 v 2 x u -4 v x u - v 2 w x u + 3 v w x u -2 w x u + 2 x u - u + v 2 - v - v 2 w + 2 v w -2 v 2 x + 2 v x + v 2 w x -2 v w x$ (db)
Jones polynomial	$q^{9/2}-4 q^{7/2}+7 q^{5/2}-12 q^{3/2}+14 \sqrt{q}-\frac{18}{\sqrt{q}}+\frac{15}{q^{3/2}}-\frac{15}{q^{5/2}}+\frac{9}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 7 z -1 -3 z a 5 - a 5 z -1 + a 5 z -3 + 3 z 3 a 3 -4 a 3 z -1 -3 a 3 z -3 - z 5 a + 2 z 3 a + 6 z a + 7 a z -1 + 3 a z -3 - z 5 a -1 - z 3 a -1 -3 z a -1 -3 a -1 z -1 - a -1 z -3 + z 3 a -3$ (db)
Kauffman polynomial	$- a 2 z 10 - z 10 -2 a 3 z 9 -6 a z 9 -4 z 9 a -1 -3 a 4 z 8 -5 a 2 z 8 -6 z 8 a -2 -8 z 8 -3 a 5 z 7 -6 a 3 z 7 + 5 a z 7 + 4 z 7 a -1 -4 z 7 a -3 -2 a 6 z 6 - a 4 z 6 + 3 a 2 z 6 + 16 z 6 a -2 - z 6 a -4 + 19 z 6 - a 7 z 5 + 3 a 5 z 5 + 19 a 3 z 5 + 13 a z 5 + 9 z 5 a -1 + 11 z 5 a -3 + 3 a 6 z 4 + 10 a 4 z 4 + 17 a 2 z 4 -10 z 4 a -2 + 2 z 4 a -4 -2 z 4 + 3 a 7 z 3 - a 5 z 3 -26 a 3 z 3 -29 a z 3 -14 z 3 a -1 -7 z 3 a -3 -16 a 4 z 2 -33 a 2 z 2 -17 z 2 -3 a 7 z + 3 a 5 z + 21 a 3 z + 28 a z + 13 z a -1 - a 6 + 11 a 4 + 24 a 2 + 13 + a 7 z -1 -3 a 5 z -1 -12 a 3 z -1 -14 a z -1 -6 a -1 z -1 -3 a 4 z -2 -6 a 2 z -2 -3 z -2 + a 5 z -3 + 3 a 3 z -3 + 3 a z -3 + a -1 z -3$ (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 L11a536. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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