L11a243

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + 3 v u 4 -2 u 4 + 4 v 2 u 3 -10 v u 3 + 5 u 3 -6 v 2 u 2 + 13 v u 2 -6 u 2 + 5 v 2 u -10 v u + 4 u -2 v 2 + 3 v -1$ (db)
Jones polynomial	$q^{9/2}-4 q^{7/2}+9 q^{5/2}-16 q^{3/2}+20 \sqrt{q}-\frac{25}{\sqrt{q}}+\frac{24}{q^{3/2}}-\frac{21}{q^{5/2}}+\frac{16}{q^{7/2}}-\frac{9}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z 7 -2 a 3 z 5 + 3 a z 5 -2 z 5 a -1 + a 5 z 3 -4 a 3 z 3 + 5 a z 3 -4 z 3 a -1 + z 3 a -3 + a 5 z -3 a 3 z + 5 a z -3 z a -1 + z a -3 - a 3 z -1 + 3 a z -1 -2 a -1 z -1$ (db)
Kauffman polynomial	$-2 a 2 z 10 -2 z 10 -7 a 3 z 9 -13 a z 9 -6 z 9 a -1 -10 a 4 z 8 -18 a 2 z 8 -7 z 8 a -2 -15 z 8 -8 a 5 z 7 - a 3 z 7 + 14 a z 7 + 3 z 7 a -1 -4 z 7 a -3 -4 a 6 z 6 + 13 a 4 z 6 + 41 a 2 z 6 + 14 z 6 a -2 - z 6 a -4 + 39 z 6 - a 7 z 5 + 11 a 5 z 5 + 16 a 3 z 5 + 10 a z 5 + 15 z 5 a -1 + 9 z 5 a -3 + 5 a 6 z 4 -6 a 4 z 4 -26 a 2 z 4 -7 z 4 a -2 + 2 z 4 a -4 -24 z 4 + a 7 z 3 -6 a 5 z 3 -11 a 3 z 3 -12 a z 3 -15 z 3 a -1 -7 z 3 a -3 -2 a 6 z 2 + 3 a 2 z 2 - z 2 a -4 + 2 z 2 + a 5 z + 2 a 3 z + 5 a z + 6 z a -1 + 2 z a -3 + a 4 + 3 a 2 + 3- a 3 z -1 -3 a z -1 -2 a -1 z -1$ (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 L11a243. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -6$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -3$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -2$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -1$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 0$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{14}$
$r = 1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 2$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{10}$
$r = 3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 4$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 5$	${\mathbb Z}_2$	${\mathbb Z}$