L11n155

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_7,16,8,17 X_13,20,14,21 X_15,22,16,7 X_6,19,1,20 X_18,11,19,12 X_12,6,13,5 X_21,14,22,15 X_4,18,5,17
Gauss code	{1, -2, 3, -11, 9, -7}, {-4, -1, 2, -3, 8, -9, -5, 10, -6, 4, 11, -8, 7, 5, -10, 6}

A Braid Representative

A Morse Link Presentation

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

-5 is the signature of L11n155. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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