L11a159

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + 2 v u 4 - u 4 + 4 v 2 u 3 -9 v u 3 + 4 u 3 -6 v 2 u 2 + 15 v u 2 -6 u 2 + 4 v 2 u -9 v u + 4 u - v 2 + 2 v -1$ (db)
Jones polynomial	$q^{9/2}-4 q^{7/2}+8 q^{5/2}-14 q^{3/2}+19 \sqrt{q}-\frac{22}{\sqrt{q}}+\frac{22}{q^{3/2}}-\frac{20}{q^{5/2}}+\frac{14}{q^{7/2}}-\frac{9}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z 7 -2 a 3 z 5 + 4 a z 5 -2 z 5 a -1 + a 5 z 3 -5 a 3 z 3 + 9 a z 3 -5 z 3 a -1 + z 3 a -3 + a 5 z -6 a 3 z + 8 a z -5 z a -1 + z a -3 + a 5 z -1 -2 a 3 z -1 + 2 a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$- a 2 z 10 - z 10 -5 a 3 z 9 -9 a z 9 -4 z 9 a -1 -9 a 4 z 8 -20 a 2 z 8 -6 z 8 a -2 -17 z 8 -8 a 5 z 7 -9 a 3 z 7 -2 a z 7 -5 z 7 a -1 -4 z 7 a -3 -4 a 6 z 6 + 12 a 4 z 6 + 46 a 2 z 6 + 12 z 6 a -2 - z 6 a -4 + 43 z 6 - a 7 z 5 + 13 a 5 z 5 + 36 a 3 z 5 + 43 a z 5 + 31 z 5 a -1 + 10 z 5 a -3 + 5 a 6 z 4 -6 a 4 z 4 -35 a 2 z 4 -6 z 4 a -2 + 2 z 4 a -4 -32 z 4 + a 7 z 3 -9 a 5 z 3 -36 a 3 z 3 -48 a z 3 -30 z 3 a -1 -8 z 3 a -3 - a 6 z 2 + 2 a 4 z 2 + 10 a 2 z 2 + z 2 a -2 - z 2 a -4 + 9 z 2 + 4 a 5 z + 14 a 3 z + 18 a z + 10 z a -1 + 2 z a -3 - a 2 - a 5 z -1 -2 a 3 z -1 -2 a z -1 - a -1 z -1$ (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 L11a159. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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