L11n157

(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_17,15,18,14 X_7,17,8,16 X_15,7,16,22 X_13,19,14,18 X_6,20,1,19 X_20,12,21,11 X_12,6,13,5 X_4,21,5,22
Gauss code	{1, -2, 3, -11, 10, -8}, {-5, -1, 2, -3, 9, -10, -7, 4, -6, 5, -4, 7, 8, -9, 11, 6}

A Braid Representative

A Morse Link Presentation

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

1 is the signature of L11n157. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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