L11a197

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_10,4,11,3 X_22,10,7,9 X_16,6,17,5 X_18,12,19,11 X_20,16,21,15 X_12,18,13,17 X_14,22,15,21 X₂₇₃₈ X_4,14,5,13 X_6,20,1,19
Gauss code	{1, -9, 2, -10, 4, -11}, {9, -1, 3, -2, 5, -7, 10, -8, 6, -4, 7, -5, 11, -6, 8, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 4 + u 4 -4 v 2 u 3 + 10 v u 3 -5 u 3 + 8 v 2 u 2 -17 v u 2 + 8 u 2 -5 v 2 u + 10 v u -4 u + v 2 - v$ (db)
Jones polynomial	$q^{21/2}-4 q^{19/2}+9 q^{17/2}-15 q^{15/2}+21 q^{13/2}-24 q^{11/2}+24 q^{9/2}-22 q^{7/2}+15 q^{5/2}-10 q^{3/2}+4 \sqrt{q}-\frac{1}{\sqrt{q}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$- z 5 a -3 -3 z 5 a -5 - z 5 a -7 + z 3 a -1 + 2 z 3 a -3 -6 z 3 a -5 + z 3 a -7 + z 3 a -9 + 5 z a -3 -6 z a -5 + 2 z a -7 + 2 a -3 z -1 -3 a -5 z -1 + a -7 z -1$ (db)
Kauffman polynomial	$-2 z 10 a -6 -2 z 10 a -8 -7 z 9 a -5 -13 z 9 a -7 -6 z 9 a -9 -11 z 8 a -4 -19 z 8 a -6 -15 z 8 a -8 -7 z 8 a -10 -9 z 7 a -3 -4 z 7 a -5 + 13 z 7 a -7 + 4 z 7 a -9 -4 z 7 a -11 -4 z 6 a -2 + 17 z 6 a -4 + 46 z 6 a -6 + 41 z 6 a -8 + 15 z 6 a -10 - z 6 a -12 - z 5 a -1 + 15 z 5 a -3 + 31 z 5 a -5 + 18 z 5 a -7 + 12 z 5 a -9 + 9 z 5 a -11 + 4 z 4 a -2 -10 z 4 a -4 -31 z 4 a -6 -29 z 4 a -8 -10 z 4 a -10 + 2 z 4 a -12 + z 3 a -1 -12 z 3 a -3 -33 z 3 a -5 -25 z 3 a -7 -11 z 3 a -9 -6 z 3 a -11 + 2 z 2 a -6 + 6 z 2 a -8 + 3 z 2 a -10 - z 2 a -12 + 7 z a -3 + 13 z a -5 + 8 z a -7 + 3 z a -9 + z a -11 + 3 a -4 + 3 a -6 + a -8 -2 a -3 z -1 -3 a -5 z -1 - a -7 z -1$ (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 L11a197. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 2$	$i = 4$
$r = -2$	${\mathbb Z}$
$r = -1$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = 0$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{4}$
$r = 1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 2$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{10}$
$r = 3$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 4$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 5$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 6$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 7$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 8$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 9$	${\mathbb Z}_2$	${\mathbb Z}$