L11n234

(Knotscape image)

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

Planar diagram presentation	X_10,1,11,2 X_5,20,6,21 X_3,15,4,14 X_15,5,16,4 X_19,22,20,9 X_7,17,8,16 X_11,18,12,19 X_17,12,18,13 X_2,9,3,10 X_13,1,14,8 X_21,6,22,7
Gauss code	{1, -9, -3, 4, -2, 11, -6, 10}, {9, -1, -7, 8, -10, 3, -4, 6, -8, 7, -5, 2, -11, 5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 3 + 2 v 2 u 3 - v u 3 + v 3 u 2 -7 v 2 u 2 + 6 v u 2 + 6 v 2 u -7 v u + u - v 2 + 2 v -1$ (db)
Jones polynomial	$-q^{7/2}+3 q^{5/2}-7 q^{3/2}+10 \sqrt{q}-\frac{12}{\sqrt{q}}+\frac{12}{q^{3/2}}-\frac{12}{q^{5/2}}+\frac{8}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{2}{q^{11/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z 7 - a 3 z 5 + 5 a z 5 - z 5 a -1 -2 a 3 z 3 + 10 a z 3 -3 z 3 a -1 - a 5 z -2 a 3 z + 7 a z -4 z a -1 + a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$- a 3 z 9 - a z 9 -2 a 4 z 8 -6 a 2 z 8 -4 z 8 - a 5 z 7 -5 a 3 z 7 -9 a z 7 -5 z 7 a -1 + 6 a 2 z 6 -3 z 6 a -2 + 3 z 6 -3 a 5 z 5 + 9 a 3 z 5 + 22 a z 5 + 9 z 5 a -1 - z 5 a -3 -3 a 6 z 4 + a 2 z 4 + 5 z 4 a -2 + 3 z 4 + 6 a 5 z 3 -5 a 3 z 3 -19 a z 3 -6 z 3 a -1 + 2 z 3 a -3 + 4 a 6 z 2 + a 4 z 2 -2 a 2 z 2 -2 z 2 a -2 - z 2 -3 a 5 z + 8 a z + 4 z a -1 - z a -3 + 1- a z -1 - a -1 z -1$ (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 L11n234. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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