L11n197

(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_5,19,6,18 X_21,16,22,17 X_15,20,16,21 X_19,14,20,15 X₂₇₃₈ X_4,11,5,12 X_17,1,18,6
Gauss code	{1, -9, 2, -10, -5, 11}, {9, -1, 3, -4, 10, -2, 4, 8, -7, 6, -11, 5, -8, 7, -6, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 4 + u 4 + v u 3 - v 2 u 2 + v u 2 - u 2 + v u + v 2 - v$ (db)
Jones polynomial	$-q^{5/2}+2 q^{3/2}-2 \sqrt{q}+\frac{2}{\sqrt{q}}-\frac{2}{q^{3/2}}-\frac{1}{q^{7/2}}-\frac{1}{q^{9/2}}+\frac{1}{q^{11/2}}-\frac{1}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- z a 7 - a 7 z -1 + z 3 a 5 + 3 z a 5 + 3 a 5 z -1 - z 3 a 3 -4 z a 3 -2 a 3 z -1 + z 5 a + 4 z 3 a + 3 z a - z 3 a -1 -2 z a -1$ (db)
Kauffman polynomial	$- a 3 z 9 - a z 9 - a 4 z 8 -3 a 2 z 8 -2 z 8 - a 7 z 7 + 6 a 3 z 7 + 4 a z 7 - z 7 a -1 - a 8 z 6 - a 6 z 6 + 7 a 4 z 6 + 18 a 2 z 6 + 11 z 6 + 5 a 7 z 5 + a 5 z 5 -9 a 3 z 5 + 5 z 5 a -1 + 5 a 8 z 4 + 6 a 6 z 4 -11 a 4 z 4 -28 a 2 z 4 -16 z 4 -6 a 7 z 3 - a 5 z 3 + 6 a 3 z 3 -5 a z 3 -6 z 3 a -1 -6 a 8 z 2 -7 a 6 z 2 + 4 a 4 z 2 + 12 a 2 z 2 + 7 z 2 + 3 a 7 z + 4 a 5 z - a 3 z + 2 z a -1 + a 8 + 3 a 6 + 3 a 4 - a 7 z -1 -3 a 5 z -1 -2 a 3 z -1$ (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 =

-3 is the signature of L11n197. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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