L11n131

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_11,19,12,18 X_3,10,4,11 X_17,3,18,2 X_5,13,6,12 X₆₇₁₈ X_16,10,17,9 X_20,14,21,13 X_22,16,7,15 X_19,4,20,5 X_14,22,15,21
Gauss code	{1, 4, -3, 10, -5, -6}, {6, -1, 7, 3, -2, 5, 8, -11, 9, -7, -4, 2, -10, -8, 11, -9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + 2 v 2 u 3 -2 v u 3 + u 3 -2 v 2 u 2 + 3 v u 2 -2 u 2 + v 2 u -2 v u + 2 u -1$ (db)
Jones polynomial	$q^{15/2}-2 q^{13/2}+4 q^{11/2}-5 q^{9/2}+6 q^{7/2}-7 q^{5/2}+5 q^{3/2}-5 \sqrt{q}+\frac{2}{\sqrt{q}}-\frac{1}{q^{3/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -3 + z 5 a -1 -6 z 5 a -3 + z 5 a -5 + 4 z 3 a -1 -13 z 3 a -3 + 4 z 3 a -5 + 6 z a -1 -12 z a -3 + 5 z a -5 + 3 a -1 z -1 -5 a -3 z -1 + 2 a -5 z -1$ (db)
Kauffman polynomial	$- z 9 a -3 - z 9 a -5 - z 8 a -2 -3 z 8 a -4 -2 z 8 a -6 + 4 z 7 a -3 + 2 z 7 a -5 -2 z 7 a -7 + 3 z 6 a -2 + 10 z 6 a -4 + 6 z 6 a -6 - z 6 a -8 -4 z 5 a -1 -13 z 5 a -3 -2 z 5 a -5 + 7 z 5 a -7 -10 z 4 a -2 -15 z 4 a -4 -3 z 4 a -6 + 4 z 4 a -8 -2 z 4 - a z 3 + 7 z 3 a -1 + 20 z 3 a -3 + 7 z 3 a -5 -5 z 3 a -7 + 11 z 2 a -2 + 14 z 2 a -4 -4 z 2 a -8 + z 2 + a z -8 z a -1 -15 z a -3 -6 z a -5 -5 a -2 -5 a -4 + a -8 + 3 a -1 z -1 + 5 a -3 z -1 + 2 a -5 z -1$ (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 L11n131. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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