L11n158

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_16,7,17,8 X_3,10,4,11 X_2,15,3,16 X_14,10,15,9 X_11,19,12,18 X_12,5,13,6 X_6,21,1,22 X_20,14,21,13 X_22,17,7,18 X_19,4,20,5
Gauss code	{1, -4, -3, 11, 7, -8}, {2, -1, 5, 3, -6, -7, 9, -5, 4, -2, 10, 6, -11, -9, 8, -10}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 4 + u 4 + v u 3 -2 u 3 - v u 2 -2 v 2 u + v u + v 2 - v$ (db)
Jones polynomial	$-\frac{1}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{3}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{2}{q^{15/2}}-\frac{1}{q^{17/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z 3 a 7 + 2 z a 7 + a 7 z -1 - z 5 a 5 -4 z 3 a 5 -5 z a 5 - a 5 z -1 + z 3 a 3 + z a 3 - z a$ (db)
Kauffman polynomial	$- z 7 a 9 + 5 z 5 a 9 -7 z 3 a 9 + 3 z a 9 -2 z 8 a 8 + 10 z 6 a 8 -13 z 4 a 8 + 4 z 2 a 8 - z 9 a 7 + 3 z 7 a 7 + 2 z 5 a 7 -5 z 3 a 7 - z a 7 + a 7 z -1 -3 z 8 a 6 + 14 z 6 a 6 -17 z 4 a 6 + 7 z 2 a 6 - a 6 - z 9 a 5 + 4 z 7 a 5 -4 z 5 a 5 + 4 z 3 a 5 -3 z a 5 + a 5 z -1 - z 8 a 4 + 4 z 6 a 4 -4 z 4 a 4 + 2 z 2 a 4 - z 5 a 3 + 2 z 3 a 3 - z 2 a 2 - z a$ (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 L11n158. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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