L11a19

(Knotscape image)

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

Visit L11a19's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_10,4,11,3 X_12,8,13,7 X_16,10,17,9 X_20,14,21,13 X_22,17,5,18 X_18,21,19,22 X_14,20,15,19 X_8,16,9,15 X₂₅₃₆ X_4,12,1,11
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 4, -2, 11, -3, 5, -8, 9, -4, 6, -7, 8, -5, 7, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + u 5 + 5 v u 4 -5 u 4 -8 v u 3 + 8 u 3 + 8 v u 2 -8 u 2 -5 v u + 5 u + v -1$ (db)
Jones polynomial	$q^{17/2}-3 q^{15/2}+7 q^{13/2}-12 q^{11/2}+16 q^{9/2}-18 q^{7/2}+17 q^{5/2}-16 q^{3/2}+11 \sqrt{q}-\frac{7}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z 7 a -3 -2 z 5 a -1 + 4 z 5 a -3 -2 z 5 a -5 + a z 3 -6 z 3 a -1 + 8 z 3 a -3 -6 z 3 a -5 + z 3 a -7 + 2 a z -6 z a -1 + 9 z a -3 -7 z a -5 + 2 z a -7 + a z -1 -2 a -1 z -1 + 3 a -3 z -1 -3 a -5 z -1 + a -7 z -1$ (db)
Kauffman polynomial	$- z 10 a -2 - z 10 a -4 -3 z 9 a -1 -8 z 9 a -3 -5 z 9 a -5 -10 z 8 a -2 -16 z 8 a -4 -9 z 8 a -6 -3 z 8 - a z 7 + 4 z 7 a -1 + 10 z 7 a -3 -4 z 7 a -5 -9 z 7 a -7 + 41 z 6 a -2 + 49 z 6 a -4 + 13 z 6 a -6 -6 z 6 a -8 + 11 z 6 + 4 a z 5 + 13 z 5 a -1 + 25 z 5 a -3 + 33 z 5 a -5 + 14 z 5 a -7 -3 z 5 a -9 -42 z 4 a -2 -40 z 4 a -4 -4 z 4 a -6 + 6 z 4 a -8 - z 4 a -10 -13 z 4 -6 a z 3 -25 z 3 a -1 -40 z 3 a -3 -36 z 3 a -5 -13 z 3 a -7 + 2 z 3 a -9 + 15 z 2 a -2 + 11 z 2 a -4 -4 z 2 a -6 -4 z 2 a -8 + z 2 a -10 + 5 z 2 + 4 a z + 13 z a -1 + 20 z a -3 + 16 z a -5 + 5 z a -7 -2 a -2 + 2 a -6 + a -8 - a z -1 -2 a -1 z -1 -3 a -3 z -1 -3 a -5 z -1 - a -7 z -1$ (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 L11a19. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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