L11n142

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_18,11,19,12 X_3,10,4,11 X_17,3,18,2 X_12,5,13,6 X₆₇₁₈ X_9,16,10,17 X_20,16,21,15 X_22,14,7,13 X_14,22,15,21 X_4,20,5,19
Gauss code	{1, 4, -3, -11, 5, -6}, {6, -1, -7, 3, 2, -5, 9, -10, 8, 7, -4, -2, 11, -8, 10, -9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v 2 u 2 + 3 v u 2 + 2 v 2 u -7 v u + 2 u + 3 v -2$ (db)
Jones polynomial	$q^{5/2}-3 q^{3/2}+4 \sqrt{q}-\frac{7}{\sqrt{q}}+\frac{7}{q^{3/2}}-\frac{7}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z 3 a 5 + 2 z a 5 + a 5 z -1 - z 5 a 3 -3 z 3 a 3 -4 z a 3 -2 a 3 z -1 - z 5 a -2 z 3 a + 2 a z -1 + z 3 a -1 + z a -1 - a -1 z -1$ (db)
Kauffman polynomial	$- a 3 z 9 - a z 9 -3 a 4 z 8 -4 a 2 z 8 - z 8 -3 a 5 z 7 + 3 a z 7 -2 a 6 z 6 + 11 a 4 z 6 + 16 a 2 z 6 + 3 z 6 - a 7 z 5 + 9 a 5 z 5 + 8 a 3 z 5 -5 a z 5 -3 z 5 a -1 + 5 a 6 z 4 -16 a 4 z 4 -27 a 2 z 4 - z 4 a -2 -7 z 4 + 3 a 7 z 3 -8 a 5 z 3 -18 a 3 z 3 - a z 3 + 6 z 3 a -1 - a 6 z 2 + 8 a 4 z 2 + 13 a 2 z 2 + 2 z 2 a -2 + 6 z 2 - a 7 z + 4 a 5 z + 10 a 3 z + 4 a z - z a -1 - a 2 - a 5 z -1 -2 a 3 z -1 -2 a z -1 - a -1 z -1$ (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 L11n142. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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