L11a232

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + 3 v u 4 -2 u 4 + 4 v 2 u 3 -11 v u 3 + 6 u 3 -7 v 2 u 2 + 17 v u 2 -7 u 2 + 6 v 2 u -11 v u + 4 u -2 v 2 + 3 v -1$ (db)
Jones polynomial	$-q^{13/2}+4 q^{11/2}-10 q^{9/2}+17 q^{7/2}-24 q^{5/2}+27 q^{3/2}-28 \sqrt{q}+\frac{24}{\sqrt{q}}-\frac{18}{q^{3/2}}+\frac{11}{q^{5/2}}-\frac{5}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + 2 a z 5 -3 z 5 a -1 + 2 z 5 a -3 - a 3 z 3 + 3 a z 3 -6 z 3 a -1 + 4 z 3 a -3 - z 3 a -5 + 2 a z -5 z a -1 + 5 z a -3 - z a -5 + a z -1 -2 a -1 z -1 + 2 a -3 z -1 - a -5 z -1$ (db)
Kauffman polynomial	$-3 z 10 a -2 -3 z 10 -9 a z 9 -18 z 9 a -1 -9 z 9 a -3 -10 a 2 z 8 -21 z 8 a -2 -12 z 8 a -4 -19 z 8 -5 a 3 z 7 + 9 a z 7 + 23 z 7 a -1 -9 z 7 a -5 - a 4 z 6 + 21 a 2 z 6 + 53 z 6 a -2 + 17 z 6 a -4 -4 z 6 a -6 + 54 z 6 + 9 a 3 z 5 + 12 a z 5 + 13 z 5 a -1 + 24 z 5 a -3 + 13 z 5 a -5 - z 5 a -7 + a 4 z 4 -12 a 2 z 4 -38 z 4 a -2 -11 z 4 a -4 + 4 z 4 a -6 -36 z 4 -4 a 3 z 3 -13 a z 3 -25 z 3 a -1 -27 z 3 a -3 -10 z 3 a -5 + z 3 a -7 + a 2 z 2 + 10 z 2 a -2 + 3 z 2 a -4 - z 2 a -6 + 7 z 2 + 4 a z + 12 z a -1 + 12 z a -3 + 4 z a -5 - a -2 - a z -1 -2 a -1 z -1 -2 a -3 z -1 - a -5 z -1$ (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 L11a232. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 0$	$i = 2$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -2$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = -1$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 0$	${\mathbb Z}^{15}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{14}$
$r = 1$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{14}$	${\mathbb Z}^{14}$
$r = 2$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{14}$
$r = 3$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 5$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 6$	${\mathbb Z}_2$	${\mathbb Z}$