L11a3

(Knotscape image)

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

Visit L11a3's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_14,10,15,9 X₈₄₉₃ X_10,5,11,6 X_20,12,21,11 X_22,20,5,19 X_18,13,19,14 X_12,22,13,21 X_2,16,3,15
Gauss code	{1, -11, 5, -3}, {6, -1, 2, -5, 4, -6, 7, -10, 9, -4, 11, -2, 3, -9, 8, -7, 10, -8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + u 5 + 6 v u 4 -6 u 4 -12 v u 3 + 12 u 3 + 12 v u 2 -12 u 2 -6 v u + 6 u + v -1$ (db)
Jones polynomial	$-q^{13/2}+5 q^{11/2}-10 q^{9/2}+16 q^{7/2}-22 q^{5/2}+24 q^{3/2}-25 \sqrt{q}+\frac{20}{\sqrt{q}}-\frac{15}{q^{3/2}}+\frac{9}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + 2 a z 5 -3 z 5 a -1 + 2 z 5 a -3 - a 3 z 3 + 4 a z 3 -4 z 3 a -1 + 3 z 3 a -3 - z 3 a -5 - a 3 z + 2 a z - z a -3 + 2 a -1 z -1 -3 a -3 z -1 + a -5 z -1$ (db)
Kauffman polynomial	$-2 z 10 a -2 -2 z 10 -6 a z 9 -13 z 9 a -1 -7 z 9 a -3 -7 a 2 z 8 -19 z 8 a -2 -11 z 8 a -4 -15 z 8 -4 a 3 z 7 + 4 a z 7 + 13 z 7 a -1 -5 z 7 a -3 -10 z 7 a -5 - a 4 z 6 + 15 a 2 z 6 + 43 z 6 a -2 + 12 z 6 a -4 -5 z 6 a -6 + 42 z 6 + 9 a 3 z 5 + 14 a z 5 + 16 z 5 a -1 + 26 z 5 a -3 + 14 z 5 a -5 - z 5 a -7 + 2 a 4 z 4 -9 a 2 z 4 -27 z 4 a -2 - z 4 a -4 + 5 z 4 a -6 -32 z 4 -6 a 3 z 3 -16 a z 3 -20 z 3 a -1 -15 z 3 a -3 -5 z 3 a -5 - a 4 z 2 + 2 a 2 z 2 + 7 z 2 a -2 + 3 z 2 a -4 + 7 z 2 + 2 a 3 z + 4 a z + z a -1 - z a -3 -3 a -2 -3 a -4 - a -6 + 2 a -1 z -1 + 3 a -3 z -1 + a -5 z -1$ (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 L11a3. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 0$	$i = 2$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -1$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 0$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{13}$
$r = 1$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 2$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 4$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 5$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 6$	${\mathbb Z}_2$	${\mathbb Z}$