L11a236

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 u 4 -5 v u 3 + 6 u 3 -5 v 2 u 2 + 9 v u 2 -5 u 2 + 6 v 2 u -5 v u -2 v 2$ (db)
Jones polynomial	$-\frac{1}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{9}{q^{9/2}}-\frac{13}{q^{11/2}}+\frac{13}{q^{13/2}}-\frac{14}{q^{15/2}}+\frac{12}{q^{17/2}}-\frac{9}{q^{19/2}}+\frac{6}{q^{21/2}}-\frac{3}{q^{23/2}}+\frac{1}{q^{25/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- a 13 z -1 + 4 z a 11 + 2 a 11 z -1 -3 z 3 a 9 - z a 9 -5 z 3 a 7 -5 z a 7 - a 7 z -1 -3 z 3 a 5 - z a 5 - z 3 a 3$ (db)
Kauffman polynomial	$- z 8 a 14 + 5 z 6 a 14 -9 z 4 a 14 + 7 z 2 a 14 -2 a 14 -3 z 9 a 13 + 15 z 7 a 13 -24 z 5 a 13 + 13 z 3 a 13 -2 z a 13 + a 13 z -1 -2 z 10 a 12 + 3 z 8 a 12 + 18 z 6 a 12 -44 z 4 a 12 + 29 z 2 a 12 -5 a 12 -10 z 9 a 11 + 42 z 7 a 11 -51 z 5 a 11 + 21 z 3 a 11 -7 z a 11 + 2 a 11 z -1 -2 z 10 a 10 -7 z 8 a 10 + 50 z 6 a 10 -64 z 4 a 10 + 25 z 2 a 10 -3 a 10 -7 z 9 a 9 + 15 z 7 a 9 + 6 z 5 a 9 -14 z 3 a 9 + 2 z a 9 -11 z 8 a 8 + 28 z 6 a 8 -15 z 4 a 8 + 2 z 2 a 8 + a 8 -12 z 7 a 7 + 27 z 5 a 7 -17 z 3 a 7 + 6 z a 7 - a 7 z -1 -9 z 6 a 6 + 11 z 4 a 6 - z 2 a 6 -6 z 5 a 5 + 4 z 3 a 5 - z a 5 -3 z 4 a 4 - z 3 a 3$ (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 L11a236. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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