L11a213

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + 3 v u 4 -2 u 4 + 4 v 2 u 3 -7 v u 3 + 5 u 3 -6 v 2 u 2 + 9 v u 2 -6 u 2 + 5 v 2 u -7 v u + 4 u -2 v 2 + 3 v -1$ (db)
Jones polynomial	$q^{17/2}-4 q^{15/2}+9 q^{13/2}-14 q^{11/2}+19 q^{9/2}-21 q^{7/2}+20 q^{5/2}-18 q^{3/2}+12 \sqrt{q}-\frac{8}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z 7 a -3 -2 z 5 a -1 + 3 z 5 a -3 -2 z 5 a -5 + a z 3 -5 z 3 a -1 + 3 z 3 a -3 -4 z 3 a -5 + z 3 a -7 + 2 a z -3 z a -1 - z a -5 + z a -7 + a z -1 -2 a -3 z -1 + a -5 z -1$ (db)
Kauffman polynomial	$-2 z 10 a -2 -2 z 10 a -4 -4 z 9 a -1 -12 z 9 a -3 -8 z 9 a -5 -6 z 8 a -2 -16 z 8 a -4 -13 z 8 a -6 -3 z 8 - a z 7 + 9 z 7 a -1 + 27 z 7 a -3 + 4 z 7 a -5 -13 z 7 a -7 + 33 z 6 a -2 + 51 z 6 a -4 + 19 z 6 a -6 -9 z 6 a -8 + 10 z 6 + 4 a z 5 + z 5 a -1 -9 z 5 a -3 + 15 z 5 a -5 + 17 z 5 a -7 -4 z 5 a -9 -35 z 4 a -2 -42 z 4 a -4 -8 z 4 a -6 + 8 z 4 a -8 - z 4 a -10 -10 z 4 -6 a z 3 -10 z 3 a -1 -3 z 3 a -3 -8 z 3 a -5 -8 z 3 a -7 + z 3 a -9 + 13 z 2 a -2 + 18 z 2 a -4 + 4 z 2 a -6 -3 z 2 a -8 + 2 z 2 + 4 a z + 3 z a -1 -2 z a -3 + z a -7 -3 a -2 -5 a -4 -2 a -6 + 1- a z -1 + 2 a -3 z -1 + a -5 z -1$ (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 =

3 is the signature of L11a213. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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