L11a221

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v 2 u 2 + 6 v u 2 -4 u 2 + 6 v 2 u -13 v u + 6 u -4 v 2 + 6 v -2$ (db)
Jones polynomial	$-q^{7/2}+3 q^{5/2}-6 q^{3/2}+10 \sqrt{q}-\frac{14}{\sqrt{q}}+\frac{15}{q^{3/2}}-\frac{16}{q^{5/2}}+\frac{13}{q^{7/2}}-\frac{10}{q^{9/2}}+\frac{6}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z a 7 + 2 z 3 a 5 + z a 5 - z 5 a 3 + a 3 z -1 - z 5 a - z a - a z -1 + 2 z 3 a -1 + z a -1 - z a -3$ (db)
Kauffman polynomial	$- a 4 z 10 - a 2 z 10 -3 a 5 z 9 -6 a 3 z 9 -3 a z 9 -4 a 6 z 8 -6 a 4 z 8 -6 a 2 z 8 -4 z 8 -3 a 7 z 7 + 3 a 5 z 7 + 11 a 3 z 7 + a z 7 -4 z 7 a -1 - a 8 z 6 + 10 a 6 z 6 + 20 a 4 z 6 + 15 a 2 z 6 -3 z 6 a -2 + 3 z 6 + 9 a 7 z 5 + 6 a 5 z 5 -8 a 3 z 5 + a z 5 + 5 z 5 a -1 - z 5 a -3 + 3 a 8 z 4 -6 a 6 z 4 -20 a 4 z 4 -15 a 2 z 4 + 6 z 4 a -2 + 2 z 4 -7 a 7 z 3 -6 a 5 z 3 + 3 a 3 z 3 + 2 z 3 a -3 -2 a 8 z 2 + 2 a 6 z 2 + 7 a 4 z 2 + 5 a 2 z 2 -3 z 2 a -2 - z 2 + 2 a 7 z + a 5 z -3 a 3 z - a z - z a -3 - a 2 + a 3 z -1 + a z -1$ (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 L11a221. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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