L11a157

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 4 + u 4 -2 v 2 u 3 + 8 v u 3 -5 u 3 + 6 v 2 u 2 -15 v u 2 + 6 u 2 -5 v 2 u + 8 v u -2 u + v 2 - v$ (db)
Jones polynomial	$q^{15/2}-4 q^{13/2}+8 q^{11/2}-13 q^{9/2}+17 q^{7/2}-20 q^{5/2}+19 q^{3/2}-17 \sqrt{q}+\frac{12}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{1}{q^{7/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z 5 a -1 + 2 z 5 a -3 -2 a z 3 -2 z 3 a -1 + 4 z 3 a -3 -3 z 3 a -5 + a 3 z -4 z a -1 + 6 z a -3 -3 z a -5 + z a -7 + a z -1 -2 a -1 z -1 + 2 a -3 z -1 - a -5 z -1$ (db)
Kauffman polynomial	$- z 10 a -2 - z 10 a -4 -4 z 9 a -1 -8 z 9 a -3 -4 z 9 a -5 -14 z 8 a -2 -14 z 8 a -4 -6 z 8 a -6 -6 z 8 -5 a z 7 -3 z 7 a -1 + 4 z 7 a -3 -2 z 7 a -5 -4 z 7 a -7 -3 a 2 z 6 + 36 z 6 a -2 + 39 z 6 a -4 + 13 z 6 a -6 - z 6 a -8 + 8 z 6 - a 3 z 5 + 7 a z 5 + 18 z 5 a -1 + 25 z 5 a -3 + 25 z 5 a -5 + 10 z 5 a -7 + 5 a 2 z 4 -34 z 4 a -2 -31 z 4 a -4 -6 z 4 a -6 + 2 z 4 a -8 -6 z 4 + 2 a 3 z 3 -4 a z 3 -24 z 3 a -1 -35 z 3 a -3 -24 z 3 a -5 -7 z 3 a -7 -2 a 2 z 2 + 11 z 2 a -2 + 8 z 2 a -4 + z 2 a -6 - z 2 a -8 + 3 z 2 - a 3 z + 3 a z + 13 z a -1 + 15 z a -3 + 8 z a -5 + 2 z a -7 - a -2 - a z -1 -2 a -1 z -1 -2 a -3 z -1 - a -5 z -1$ (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 L11a157. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 0$	$i = 2$
$r = -4$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -2$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -1$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 0$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{8}$
$r = 1$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 2$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{11}$
$r = 3$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 4$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 5$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 6$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 7$	${\mathbb Z}_2$	${\mathbb Z}$