L11a181

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_20,9,21,10 X_14,5,15,6 X_16,8,17,7 X_10,4,11,3 X_22,14,7,13 X_18,12,19,11 X_12,18,13,17 X_6,15,1,16 X_4,21,5,22 X_2,20,3,19
Gauss code	{1, -11, 5, -10, 3, -9}, {4, -1, 2, -5, 7, -8, 6, -3, 9, -4, 8, -7, 11, -2, 10, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v 2 u 4 + 2 v u 4 + 7 v 2 u 3 -9 v u 3 + 3 u 3 -8 v 2 u 2 + 15 v u 2 -8 u 2 + 3 v 2 u -9 v u + 7 u + 2 v -2$ (db)
Jones polynomial	$-q^{11/2}+5 q^{9/2}-10 q^{7/2}+16 q^{5/2}-22 q^{3/2}+24 \sqrt{q}-\frac{25}{\sqrt{q}}+\frac{21}{q^{3/2}}-\frac{16}{q^{5/2}}+\frac{9}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{1}{q^{11/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z 7 + z 7 a -1 - a 3 z 5 + 3 a z 5 + 2 z 5 a -1 - z 5 a -3 -2 a 3 z 3 + 4 a z 3 -2 z 3 a -1 - z 3 a -3 - a 3 z + 4 a z -6 z a -1 + 2 z a -3 + 2 a z -1 -3 a -1 z -1 + a -3 z -1$ (db)
Kauffman polynomial	$-4 z 10 a -2 -4 z 10 -12 a z 9 -20 z 9 a -1 -8 z 9 a -3 -17 a 2 z 8 -4 z 8 a -2 -5 z 8 a -4 -16 z 8 -15 a 3 z 7 + 12 a z 7 + 52 z 7 a -1 + 24 z 7 a -3 - z 7 a -5 -9 a 4 z 6 + 29 a 2 z 6 + 39 z 6 a -2 + 15 z 6 a -4 + 62 z 6 -4 a 5 z 5 + 21 a 3 z 5 + 18 a z 5 -28 z 5 a -1 -19 z 5 a -3 + 2 z 5 a -5 - a 6 z 4 + 6 a 4 z 4 -15 a 2 z 4 -36 z 4 a -2 -12 z 4 a -4 -46 z 4 + a 5 z 3 -12 a 3 z 3 -22 a z 3 -7 z 3 a -1 + z 3 a -3 - z 3 a -5 - a 4 z 2 + 2 a 2 z 2 + 3 z 2 a -2 + 6 z 2 + 2 a 3 z + 9 a z + 10 z a -1 + 3 z a -3 + 3 a -2 + a -4 + 3-2 a z -1 -3 a -1 z -1 - a -3 z -1$ (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 L11a181. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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