L11n112

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_7,18,8,19 X_19,22,20,5 X_9,21,10,20 X_21,9,22,8 X_11,17,12,16 X_17,15,18,14 X_15,11,16,10 X₂₅₃₆ X_4,14,1,13
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 $u$ , $v$ , $w$ , ...)	$-2 v u + 2 u + 2 v -2$ (db)
Jones polynomial	$q^{15/2}-2 q^{13/2}+3 q^{11/2}-4 q^{9/2}+3 q^{7/2}-3 q^{5/2}+2 q^{3/2}-\sqrt{q}-\frac{1}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 5 a -1 + z 5 a -3 + a z 3 -6 z 3 a -1 + 5 z 3 a -3 -2 z 3 a -5 + 3 a z -9 z a -1 + 10 z a -3 -5 z a -5 + z a -7 + 2 a z -1 -5 a -1 z -1 + 6 a -3 z -1 -4 a -5 z -1 + a -7 z -1$ (db)
Kauffman polynomial	$- z 8 a -2 - z 8 a -4 - z 8 a -6 - z 8 - a z 7 -2 z 7 a -1 -3 z 7 a -3 -4 z 7 a -5 -2 z 7 a -7 + 7 z 6 a -2 + 3 z 6 a -4 + z 6 a -6 - z 6 a -8 + 6 z 6 + 6 a z 5 + 16 z 5 a -1 + 20 z 5 a -3 + 18 z 5 a -5 + 8 z 5 a -7 -10 z 4 a -2 + 2 z 4 a -4 + 8 z 4 a -6 + 4 z 4 a -8 -8 z 4 -10 a z 3 -33 z 3 a -1 -39 z 3 a -3 -24 z 3 a -5 -8 z 3 a -7 -7 z 2 a -4 -9 z 2 a -6 -4 z 2 a -8 + 2 z 2 + 7 a z + 22 z a -1 + 27 z a -3 + 16 z a -5 + 4 z a -7 + a -2 + 3 a -4 + 3 a -6 + a -8 + 1-2 a z -1 -5 a -1 z -1 -6 a -3 z -1 -4 a -5 z -1 - a -7 z -1$ (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 L11n112. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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