L11a231

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{2 u^2 v^3-7 u^2 v^2+6 u^2 v-2 u^2+u v^4-8 u v^3+13 u v^2-8 u v+u-2 v^4+6 v^3-7 v^2+2 v}{u v^2}$ (db)
Jones polynomial	$-\sqrt{q}+\frac{4}{\sqrt{q}}-\frac{9}{q^{3/2}}+\frac{14}{q^{5/2}}-\frac{19}{q^{7/2}}+\frac{21}{q^{9/2}}-\frac{21}{q^{11/2}}+\frac{17}{q^{13/2}}-\frac{13}{q^{15/2}}+\frac{7}{q^{17/2}}-\frac{3}{q^{19/2}}+\frac{1}{q^{21/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- a 11 z -1 + 4 z a 9 + 3 a 9 z -1 -5 z 3 a 7 -6 z a 7 -2 a 7 z -1 + 2 z 5 a 5 + 2 z 3 a 5 + z a 5 + z 5 a 3 - z 3 a 3 -2 z a 3 - z 3 a$ (db)
Kauffman polynomial	$- z 6 a 12 + 3 z 4 a 12 -3 z 2 a 12 + a 12 -3 z 7 a 11 + 8 z 5 a 11 -7 z 3 a 11 + 3 z a 11 - a 11 z -1 -4 z 8 a 10 + 5 z 6 a 10 + 6 z 4 a 10 -9 z 2 a 10 + 3 a 10 -4 z 9 a 9 + 2 z 7 a 9 + 11 z 5 a 9 -15 z 3 a 9 + 12 z a 9 -3 a 9 z -1 -2 z 10 a 8 -6 z 8 a 8 + 17 z 6 a 8 -8 z 4 a 8 -3 z 2 a 8 + 3 a 8 -11 z 9 a 7 + 20 z 7 a 7 -9 z 5 a 7 -6 z 3 a 7 + 7 z a 7 -2 a 7 z -1 -2 z 10 a 6 -12 z 8 a 6 + 34 z 6 a 6 -31 z 4 a 6 + 8 z 2 a 6 -7 z 9 a 5 + 7 z 7 a 5 + 2 z 5 a 5 -5 z 3 a 5 -10 z 8 a 4 + 19 z 6 a 4 -15 z 4 a 4 + 5 z 2 a 4 -8 z 7 a 3 + 13 z 5 a 3 -6 z 3 a 3 + 2 z a 3 -4 z 6 a 2 + 5 z 4 a 2 - z 5 a + z 3 a$ (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 L11a231. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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