L11a220

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + 2 v u 4 - u 4 + 4 v 2 u 3 -8 v u 3 + 5 u 3 -7 v 2 u 2 + 13 v u 2 -7 u 2 + 5 v 2 u -8 v u + 4 u - v 2 + 2 v -1$ (db)
Jones polynomial	$-q^{7/2}+4 q^{5/2}-9 q^{3/2}+15 \sqrt{q}-\frac{20}{\sqrt{q}}+\frac{22}{q^{3/2}}-\frac{23}{q^{5/2}}+\frac{18}{q^{7/2}}-\frac{14}{q^{9/2}}+\frac{8}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z a 7 - a 7 z -1 + 3 z 3 a 5 + 5 z a 5 + 3 a 5 z -1 -3 z 5 a 3 -8 z 3 a 3 -8 z a 3 -2 a 3 z -1 + z 7 a + 4 z 5 a + 8 z 3 a + 5 z a - z 5 a -1 -2 z 3 a -1 -2 z a -1$ (db)
Kauffman polynomial	$- a 4 z 10 - a 2 z 10 -4 a 5 z 9 -9 a 3 z 9 -5 a z 9 -5 a 6 z 8 -16 a 4 z 8 -20 a 2 z 8 -9 z 8 -3 a 7 z 7 -2 a 5 z 7 - a 3 z 7 -10 a z 7 -8 z 7 a -1 - a 8 z 6 + 9 a 6 z 6 + 39 a 4 z 6 + 43 a 2 z 6 -4 z 6 a -2 + 10 z 6 + 7 a 7 z 5 + 20 a 5 z 5 + 35 a 3 z 5 + 35 a z 5 + 12 z 5 a -1 - z 5 a -3 + 3 a 8 z 4 -3 a 6 z 4 -27 a 4 z 4 -28 a 2 z 4 + 5 z 4 a -2 -2 z 4 -6 a 7 z 3 -20 a 5 z 3 -33 a 3 z 3 -28 a z 3 -8 z 3 a -1 + z 3 a -3 -3 a 8 z 2 -3 a 6 z 2 + 4 a 4 z 2 + 6 a 2 z 2 -2 z 2 a -2 + 3 a 7 z + 9 a 5 z + 10 a 3 z + 7 a z + 3 z a -1 + a 8 + 3 a 6 + 3 a 4 - a 7 z -1 -3 a 5 z -1 -2 a 3 z -1$ (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 L11a220. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -7$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -4$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -3$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = -2$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{11}$
$r = -1$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 0$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{11}$
$r = 1$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 2$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 4$	${\mathbb Z}_2$	${\mathbb Z}$