L11n122

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_14,3,15,4 X_16,10,17,9 X_11,21,12,20 X_21,9,22,8 X_7,19,8,18 X_19,13,20,12 X_10,16,11,15 X_22,17,5,18 X₂₅₃₆ X_4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, -6, 5, 3, -8, -4, 7, 11, -2, 8, -3, 9, 6, -7, 4, -5, -9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$u 3 -2 v u 2 + 2 u 2 + 2 v u -2 u + v$ (db)
Jones polynomial	$-q^{11/2}+2 q^{9/2}-2 q^{7/2}+3 q^{5/2}-2 q^{3/2}+\sqrt{q}-\frac{2}{\sqrt{q}}-\frac{1}{q^{7/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z a 3 + 2 a 3 z -1 - z 3 a -5 z a -3 a z -1 + z 3 a -1 + 2 z a -1 + a -1 z -1 + z 3 a -3 + z a -3 - z a -5$ (db)
Kauffman polynomial	$- z 9 a -1 - z 9 a -3 -3 z 8 a -2 -2 z 8 a -4 - z 8 + 5 z 7 a -1 + 4 z 7 a -3 - z 7 a -5 + 17 z 6 a -2 + 11 z 6 a -4 + 6 z 6 - a 3 z 5 -5 z 5 a -1 - z 5 a -3 + 5 z 5 a -5 -26 z 4 a -2 -17 z 4 a -4 -9 z 4 + 5 a 3 z 3 + 4 a z 3 + 2 z 3 a -1 -3 z 3 a -3 -6 z 3 a -5 + 3 a 2 z 2 + 13 z 2 a -2 + 9 z 2 a -4 + 7 z 2 -5 a 3 z -7 a z -3 z a -1 + z a -5 -3 a 2 - a -2 -3 + 2 a 3 z -1 + 3 a z -1 + a -1 z -1$ (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 L11n122. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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