L11n129

(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_4,15,5,16 X_5,13,6,12 X_16,12,17,11 X_17,6,18,1 X_19,14,20,15 X_13,20,14,21 X_21,19,22,18
Gauss code	{1, -4, 2, -5, -6, 8}, {4, -1, 3, -2, 7, 6, -10, 9, 5, -7, -8, 11, -9, 10, -11, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 2 + 3 v u 2 - u 2 + 2 v 2 u -7 v u + 2 u - v 2 + 3 v -1$ (db)
Jones polynomial	$q^{9/2}-3 q^{7/2}+4 q^{5/2}-6 q^{3/2}+7 \sqrt{q}-\frac{7}{\sqrt{q}}+\frac{6}{q^{3/2}}-\frac{5}{q^{5/2}}+\frac{2}{q^{7/2}}-\frac{1}{q^{9/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 5 z -1 -3 z a 3 -2 a 3 z -1 + 3 z 3 a + 5 z a + 2 a z -1 - z 5 a -1 -3 z 3 a -1 -4 z a -1 - a -1 z -1 + z 3 a -3 + z a -3$ (db)
Kauffman polynomial	$- a z 9 - z 9 a -1 -2 a 2 z 8 -3 z 8 a -2 -5 z 8 - a 3 z 7 -2 z 7 a -1 -3 z 7 a -3 + 8 a 2 z 6 + 10 z 6 a -2 - z 6 a -4 + 19 z 6 + 3 a 3 z 5 + 10 a z 5 + 18 z 5 a -1 + 11 z 5 a -3 -2 a 4 z 4 -15 a 2 z 4 -7 z 4 a -2 + 3 z 4 a -4 -23 z 4 - a 5 z 3 -9 a 3 z 3 -21 a z 3 -22 z 3 a -1 -9 z 3 a -3 + 2 a 4 z 2 + 8 a 2 z 2 + 2 z 2 a -2 - z 2 a -4 + 9 z 2 + 2 a 5 z + 8 a 3 z + 12 a z + 8 z a -1 + 2 z a -3 - a 2 - a 5 z -1 -2 a 3 z -1 -2 a z -1 - a -1 z -1$ (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 L11n129. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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