L11a188

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 6 + u 6 - v 2 u 5 + 3 v u 5 - u 5 + v 2 u 4 -3 v u 4 + u 4 - v 2 u 3 + 3 v u 3 - u 3 + v 2 u 2 -3 v u 2 + u 2 - v 2 u + 3 v u - u + v 2 - v$ (db)
Jones polynomial	$-\frac{1}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{4}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{8}{q^{11/2}}-\frac{9}{q^{13/2}}+\frac{8}{q^{15/2}}-\frac{6}{q^{17/2}}+\frac{4}{q^{19/2}}-\frac{2}{q^{21/2}}+\frac{1}{q^{23/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$- z 5 a 9 -4 z 3 a 9 -4 z a 9 -2 a 9 z -1 + z 7 a 7 + 5 z 5 a 7 + 9 z 3 a 7 + 10 z a 7 + 5 a 7 z -1 + z 7 a 5 + 4 z 5 a 5 + 2 z 3 a 5 -4 z a 5 -3 a 5 z -1 - z 5 a 3 -4 z 3 a 3 -3 z a 3$ (db)
Kauffman polynomial	$- z 4 a 14 + 2 z 2 a 14 -2 z 5 a 13 + 3 z 3 a 13 -3 z 6 a 12 + 6 z 4 a 12 -5 z 2 a 12 + a 12 -3 z 7 a 11 + 6 z 5 a 11 -6 z 3 a 11 -3 z 8 a 10 + 8 z 6 a 10 -10 z 4 a 10 + 2 z 2 a 10 -3 z 9 a 9 + 13 z 7 a 9 -28 z 5 a 9 + 28 z 3 a 9 -12 z a 9 + 2 a 9 z -1 - z 10 a 8 + 11 z 6 a 8 -25 z 4 a 8 + 22 z 2 a 8 -5 a 8 -5 z 9 a 7 + 25 z 7 a 7 -50 z 5 a 7 + 53 z 3 a 7 -25 z a 7 + 5 a 7 z -1 - z 10 a 6 + z 8 a 6 + 9 z 6 a 6 -18 z 4 a 6 + 15 z 2 a 6 -5 a 6 -2 z 9 a 5 + 8 z 7 a 5 -9 z 5 a 5 + 9 z 3 a 5 -10 z a 5 + 3 a 5 z -1 -2 z 8 a 4 + 9 z 6 a 4 -10 z 4 a 4 + 2 z 2 a 4 - z 7 a 3 + 5 z 5 a 3 -7 z 3 a 3 + 3 z a 3$ (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 =

-5 is the signature of L11a188. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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