L11a119

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 3 + 4 u 3 + 9 v u 2 -12 u 2 -12 v u + 9 u + 4 v -2$ (db)
Jones polynomial	$q^{9/2}-4 q^{7/2}+7 q^{5/2}-12 q^{3/2}+16 \sqrt{q}-\frac{17}{\sqrt{q}}+\frac{17}{q^{3/2}}-\frac{15}{q^{5/2}}+\frac{10}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 7 z -1 -3 z a 5 - a 5 z -1 + 3 z 3 a 3 - a 3 z -1 - z 5 a + 2 z 3 a + 4 z a + 2 a z -1 - z 5 a -1 - z 3 a -1 -3 z a -1 - a -1 z -1 + z 3 a -3$ (db)
Kauffman polynomial	$- a 2 z 10 - z 10 -3 a 3 z 9 -7 a z 9 -4 z 9 a -1 -4 a 4 z 8 -9 a 2 z 8 -6 z 8 a -2 -11 z 8 -3 a 5 z 7 - a 3 z 7 + 8 a z 7 + 2 z 7 a -1 -4 z 7 a -3 -2 a 6 z 6 + 5 a 4 z 6 + 25 a 2 z 6 + 16 z 6 a -2 - z 6 a -4 + 35 z 6 - a 7 z 5 + 2 a 5 z 5 + 8 a 3 z 5 + 10 a z 5 + 16 z 5 a -1 + 11 z 5 a -3 + 3 a 6 z 4 -6 a 4 z 4 -30 a 2 z 4 -12 z 4 a -2 + 2 z 4 a -4 -35 z 4 + 3 a 7 z 3 + 3 a 5 z 3 -12 a 3 z 3 -23 a z 3 -18 z 3 a -1 -7 z 3 a -3 + 5 a 4 z 2 + 16 a 2 z 2 + 4 z 2 a -2 + 15 z 2 -3 a 7 z -3 a 5 z + 7 a 3 z + 13 a z + 6 z a -1 - a 6 -2 a 4 -3 a 2 -1 + a 7 z -1 + a 5 z -1 - a 3 z -1 -2 a z -1 - a -1 z -1$ (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 L11a119. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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