L11a214

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + 3 v u 4 + 4 v 2 u 3 -9 v u 3 + 4 u 3 -6 v 2 u 2 + 13 v u 2 -6 u 2 + 4 v 2 u -9 v u + 4 u + 3 v -1$ (db)
Jones polynomial	$q^{9/2}-4 q^{7/2}+9 q^{5/2}-14 q^{3/2}+19 \sqrt{q}-\frac{22}{\sqrt{q}}+\frac{21}{q^{3/2}}-\frac{19}{q^{5/2}}+\frac{13}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z 7 -2 a 3 z 5 + 3 a z 5 -2 z 5 a -1 + a 5 z 3 -5 a 3 z 3 + 3 a z 3 -4 z 3 a -1 + z 3 a -3 + 2 a 5 z -4 a 3 z - a z - z a -1 + z a -3 + a 5 z -1 -2 a z -1 + a -1 z -1$ (db)
Kauffman polynomial	$-2 a 2 z 10 -2 z 10 -6 a 3 z 9 -12 a z 9 -6 z 9 a -1 -7 a 4 z 8 -12 a 2 z 8 -7 z 8 a -2 -12 z 8 -6 a 5 z 7 + 5 a 3 z 7 + 22 a z 7 + 7 z 7 a -1 -4 z 7 a -3 -3 a 6 z 6 + 8 a 4 z 6 + 34 a 2 z 6 + 16 z 6 a -2 - z 6 a -4 + 40 z 6 - a 7 z 5 + 10 a 5 z 5 - a 3 z 5 -14 a z 5 + 7 z 5 a -1 + 9 z 5 a -3 + 4 a 6 z 4 -2 a 4 z 4 -35 a 2 z 4 -10 z 4 a -2 + 2 z 4 a -4 -41 z 4 + 2 a 7 z 3 -9 a 5 z 3 + 9 a z 3 -7 z 3 a -1 -5 z 3 a -3 - a 6 z 2 -2 a 4 z 2 + 16 a 2 z 2 + 4 z 2 a -2 - z 2 a -4 + 22 z 2 - a 7 z + 5 a 5 z -7 a z + z a -3 + a 4 -3 a 2 -2 a -2 -5- a 5 z -1 + 2 a z -1 + a -1 z -1$ (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 L11a214. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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