L11n111

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_3,12,4,13 X_7,18,8,19 X_19,22,20,5 X_9,21,10,20 X_21,9,22,8 X_16,11,17,12 X_14,17,15,18 X_10,15,11,16 X₂₅₃₆ X_13,4,14,1
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 6, -5, -9, 7, 2, -11, -8, 9, -7, 8, 3, -4, 5, -6, 4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + 2 v u 4 + u 4 + v u 3 -2 u 3 -2 v u 2 + u 2 + v u + 2 u -1$ (db)
Jones polynomial	$-\frac{1}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{2}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{1}{q^{11/2}}+\frac{1}{q^{13/2}}-\frac{2}{q^{15/2}}+\frac{2}{q^{17/2}}-\frac{2}{q^{19/2}}+\frac{1}{q^{21/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- a 11 z -1 + 4 z a 9 + 4 a 9 z -1 - z 5 a 7 -7 z 3 a 7 -11 z a 7 -6 a 7 z -1 + z 7 a 5 + 6 z 5 a 5 + 11 z 3 a 5 + 11 z a 5 + 5 a 5 z -1 - z 5 a 3 -5 z 3 a 3 -6 z a 3 -2 a 3 z -1$ (db)
Kauffman polynomial	$- z 6 a 12 + 4 z 4 a 12 -3 z 2 a 12 + a 12 -2 z 7 a 11 + 9 z 5 a 11 -9 z 3 a 11 + 4 z a 11 - a 11 z -1 - z 8 a 10 + 3 z 6 a 10 + 4 z 4 a 10 -9 z 2 a 10 + 3 a 10 -3 z 7 a 9 + 18 z 5 a 9 -29 z 3 a 9 + 18 z a 9 -4 a 9 z -1 - z 8 a 8 + 5 z 6 a 8 - z 4 a 8 -7 z 2 a 8 + 3 a 8 -3 z 7 a 7 + 22 z 5 a 7 -42 z 3 a 7 + 28 z a 7 -6 a 7 z -1 - z 8 a 6 + 6 z 6 a 6 -6 z 4 a 6 - z 2 a 6 + a 6 -3 z 7 a 5 + 19 z 5 a 5 -33 z 3 a 5 + 22 z a 5 -5 a 5 z -1 - z 8 a 4 + 5 z 6 a 4 -5 z 4 a 4 + a 4 - z 7 a 3 + 6 z 5 a 3 -11 z 3 a 3 + 8 z a 3 -2 a 3 z -1$ (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 =

-3 is the signature of L11n111. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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