L11a227

(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_14,18,15,17 X_6,15,1,16 X_22,14,7,13 X_18,22,19,21 X_2,11,3,12 X_12,3,13,4 X_10,19,11,20
Gauss code	{1, -9, 10, -3, 4, -6}, {3, -1, 2, -11, 9, -10, 7, -5, 6, -4, 5, -8, 11, -2, 8, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + 2 v u 4 + 5 v 2 u 3 -10 v u 3 + 3 u 3 -8 v 2 u 2 + 15 v u 2 -8 u 2 + 3 v 2 u -10 v u + 5 u + 2 v -1$ (db)
Jones polynomial	$q^{3/2}-5 \sqrt{q}+\frac{10}{\sqrt{q}}-\frac{16}{q^{3/2}}+\frac{21}{q^{5/2}}-\frac{24}{q^{7/2}}+\frac{23}{q^{9/2}}-\frac{20}{q^{11/2}}+\frac{14}{q^{13/2}}-\frac{8}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z a 9 + a 9 z -1 -3 z 3 a 7 -5 z a 7 -2 a 7 z -1 + 3 z 5 a 5 + 7 z 3 a 5 + 5 z a 5 + 2 a 5 z -1 - z 7 a 3 -3 z 5 a 3 -4 z 3 a 3 -3 z a 3 - a 3 z -1 + z 5 a + z 3 a - z a$ (db)
Kauffman polynomial	$- z 5 a 11 + 2 z 3 a 11 - z a 11 -3 z 6 a 10 + 4 z 4 a 10 - z 2 a 10 -6 z 7 a 9 + 9 z 5 a 9 -8 z 3 a 9 + 5 z a 9 - a 9 z -1 -8 z 8 a 8 + 12 z 6 a 8 -11 z 4 a 8 + 4 z 2 a 8 -6 z 9 a 7 - z 7 a 7 + 19 z 5 a 7 -26 z 3 a 7 + 12 z a 7 -2 a 7 z -1 -2 z 10 a 6 -16 z 8 a 6 + 42 z 6 a 6 -37 z 4 a 6 + 12 z 2 a 6 - a 6 -13 z 9 a 5 + 15 z 7 a 5 + 15 z 5 a 5 -26 z 3 a 5 + 12 z a 5 -2 a 5 z -1 -2 z 10 a 4 -17 z 8 a 4 + 48 z 6 a 4 -34 z 4 a 4 + 8 z 2 a 4 -7 z 9 a 3 + 5 z 7 a 3 + 16 z 5 a 3 -14 z 3 a 3 + 5 z a 3 - a 3 z -1 -9 z 8 a 2 + 20 z 6 a 2 -11 z 4 a 2 + z 2 a 2 -5 z 7 a + 10 z 5 a -4 z 3 a - z a - z 6 + z 4$ (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 L11a227. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -4$	$i = -2$
$r = -8$	${\mathbb Z}$
$r = -7$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{3}$
$r = -5$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -4$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = -3$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = -2$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = -1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 0$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{10}$
$r = 1$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 2$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 3$	${\mathbb Z}_2$	${\mathbb Z}$