L11a129

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 3 + 3 u 3 + 6 v u 2 -6 u 2 -6 v u + 6 u + 3 v -2$ (db)
Jones polynomial	$-q^{5/2}+3 q^{3/2}-5 \sqrt{q}+\frac{7}{\sqrt{q}}-\frac{9}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{11}{q^{7/2}}+\frac{8}{q^{9/2}}-\frac{7}{q^{11/2}}+\frac{4}{q^{13/2}}-\frac{2}{q^{15/2}}+\frac{1}{q^{17/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- a 9 z -1 + 3 z a 7 + 3 a 7 z -1 -3 z 3 a 5 -5 z a 5 -2 a 5 z -1 + z 5 a 3 + z 3 a 3 + z 5 a + 2 z 3 a + z a - z 3 a -1 - z a -1$ (db)
Kauffman polynomial	$- z 4 a 10 + 2 z 2 a 10 - a 10 -2 z 5 a 9 + 3 z 3 a 9 -2 z a 9 + a 9 z -1 -2 z 6 a 8 - z 4 a 8 + 5 z 2 a 8 -3 a 8 -2 z 7 a 7 -2 z 5 a 7 + 7 z 3 a 7 -7 z a 7 + 3 a 7 z -1 -2 z 8 a 6 + z 4 a 6 + 3 z 2 a 6 -3 a 6 -2 z 9 a 5 + 3 z 7 a 5 -4 z 5 a 5 + 7 z 3 a 5 -5 z a 5 + 2 a 5 z -1 - z 10 a 4 - z 8 a 4 + 8 z 6 a 4 -7 z 4 a 4 + 2 z 2 a 4 -5 z 9 a 3 + 17 z 7 a 3 -16 z 5 a 3 + 6 z 3 a 3 - z a 3 - z 10 a 2 -2 z 8 a 2 + 19 z 6 a 2 -23 z 4 a 2 + 7 z 2 a 2 -3 z 9 a + 11 z 7 a -8 z 5 a - z 3 a -3 z 8 + 13 z 6 -15 z 4 + 5 z 2 - z 7 a -1 + 4 z 5 a -1 -4 z 3 a -1 + z a -1$ (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 =

-3 is the signature of L11a129. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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