L11a226

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_20,9,21,10 X₄₇₅₈ X_16,5,17,6 X_6,15,1,16 X_22,17,7,18 X_18,13,19,14 X_14,21,15,22 X_2,11,3,12 X_12,3,13,4 X_10,19,11,20
Gauss code	{1, -9, 10, -3, 4, -5}, {3, -1, 2, -11, 9, -10, 7, -8, 5, -4, 6, -7, 11, -2, 8, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-3 v u 4 + u 4 -5 v 2 u 3 + 11 v u 3 -5 u 3 + 8 v 2 u 2 -17 v u 2 + 8 u 2 -5 v 2 u + 11 v u -5 u + v 2 -3 v$ (db)
Jones polynomial	$-\frac{1}{q^{5/2}}+\frac{4}{q^{7/2}}-\frac{11}{q^{9/2}}+\frac{17}{q^{11/2}}-\frac{24}{q^{13/2}}+\frac{27}{q^{15/2}}-\frac{27}{q^{17/2}}+\frac{23}{q^{19/2}}-\frac{17}{q^{21/2}}+\frac{10}{q^{23/2}}-\frac{4}{q^{25/2}}+\frac{1}{q^{27/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$- z a 13 - a 13 z -1 + 4 z 3 a 11 + 6 z a 11 + 2 a 11 z -1 -3 z 5 a 9 -5 z 3 a 9 - z a 9 -4 z 5 a 7 -10 z 3 a 7 -7 z a 7 - a 7 z -1 - z 5 a 5 - z 3 a 5$ (db)
Kauffman polynomial	$- z 6 a 16 + 2 z 4 a 16 - z 2 a 16 -4 z 7 a 15 + 8 z 5 a 15 -5 z 3 a 15 + z a 15 -8 z 8 a 14 + 17 z 6 a 14 -13 z 4 a 14 + 6 z 2 a 14 -2 a 14 -8 z 9 a 13 + 10 z 7 a 13 + 3 z 5 a 13 -4 z 3 a 13 - z a 13 + a 13 z -1 -3 z 10 a 12 -16 z 8 a 12 + 51 z 6 a 12 -49 z 4 a 12 + 25 z 2 a 12 -5 a 12 -18 z 9 a 11 + 30 z 7 a 11 -13 z 5 a 11 + 10 z 3 a 11 -9 z a 11 + 2 a 11 z -1 -3 z 10 a 10 -21 z 8 a 10 + 56 z 6 a 10 -46 z 4 a 10 + 17 z 2 a 10 -3 a 10 -10 z 9 a 9 + 6 z 7 a 9 + 9 z 5 a 9 -6 z 3 a 9 -13 z 8 a 8 + 19 z 6 a 8 -9 z 4 a 8 - z 2 a 8 + a 8 -10 z 7 a 7 + 16 z 5 a 7 -14 z 3 a 7 + 7 z a 7 - a 7 z -1 -4 z 6 a 6 + 3 z 4 a 6 - z 5 a 5 + z 3 a 5$ (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 =

-5 is the signature of L11a226. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -6$	$i = -4$
$r = -11$	${\mathbb Z}$
$r = -10$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -9$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -8$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = -7$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -6$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{14}$
$r = -5$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{13}$
$r = -4$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{14}$	${\mathbb Z}^{14}$
$r = -3$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -2$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = -1$	${\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 0$	${\mathbb Z}$	${\mathbb Z}$