L11n189

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_12,3,13,4 X_22,10,7,9 X_10,14,11,13 X_5,16,6,17 X_15,21,16,20 X_21,19,22,18 X_19,15,20,14 X₂₇₃₈ X_4,11,5,12 X_17,6,18,1
Gauss code	{1, -9, 2, -10, -5, 11}, {9, -1, 3, -4, 10, -2, 4, 8, -6, 5, -11, 7, -8, 6, -7, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- u 4 - v 2 u 3 + 2 v u 3 - u 3 + 2 v 2 u 2 -5 v u 2 + 2 u 2 - v 2 u + 2 v u - u - v 2$ (db)
Jones polynomial	$q^{9/2}-3 q^{7/2}+4 q^{5/2}-5 q^{3/2}+6 \sqrt{q}-\frac{5}{\sqrt{q}}+\frac{4}{q^{3/2}}-\frac{3}{q^{5/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z a 5 + 2 a 5 z -1 - z 3 a 3 -5 z a 3 -3 a 3 z -1 + 2 z 3 a + 3 z a + a z -1 - z 5 a -1 -3 z 3 a -1 -3 z a -1 + z 3 a -3 + z a -3$ (db)
Kauffman polynomial	$- a z 9 - z 9 a -1 - a 2 z 8 -3 z 8 a -2 -4 z 8 - a 5 z 7 + 4 a z 7 -3 z 7 a -3 + 6 a 2 z 6 + 11 z 6 a -2 - z 6 a -4 + 18 z 6 + 7 a 5 z 5 + 3 a 3 z 5 -4 a z 5 + 11 z 5 a -1 + 11 z 5 a -3 + 3 a 4 z 4 -10 a 2 z 4 -9 z 4 a -2 + 3 z 4 a -4 -25 z 4 -14 a 5 z 3 -11 a 3 z 3 - a z 3 -12 z 3 a -1 -8 z 3 a -3 -7 a 4 z 2 + 3 z 2 a -2 - z 2 a -4 + 11 z 2 + 10 a 5 z + 10 a 3 z + a z + 2 z a -1 + z a -3 + 3 a 4 + 3 a 2 + 1-2 a 5 z -1 -3 a 3 z -1 - a z -1$ (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 L11n189. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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