L11a4

(Knotscape image)

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

Visit L11a4's page at the original Knot Atlas.

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 5 + 2 u 5 + 6 v u 4 -6 u 4 -9 v u 3 + 9 u 3 + 9 v u 2 -9 u 2 -6 v u + 6 u + 2 v -2$ (db)
Jones polynomial	$-q^{19/2}+4 q^{17/2}-9 q^{15/2}+15 q^{13/2}-19 q^{11/2}+22 q^{9/2}-22 q^{7/2}+18 q^{5/2}-14 q^{3/2}+7 \sqrt{q}-\frac{4}{\sqrt{q}}+\frac{1}{q^{3/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z 7 a -3 + z 7 a -5 - z 5 a -1 + 3 z 5 a -3 + 3 z 5 a -5 - z 5 a -7 -2 z 3 a -1 + 2 z 3 a -3 + 3 z 3 a -5 -2 z 3 a -7 + z a -1 -2 z a -3 + 2 z a -5 - z a -7 + 2 a -1 z -1 -3 a -3 z -1 + a -5 z -1$ (db)
Kauffman polynomial	$-2 z 10 a -4 -2 z 10 a -6 -5 z 9 a -3 -11 z 9 a -5 -6 z 9 a -7 -6 z 8 a -2 -8 z 8 a -4 -11 z 8 a -6 -9 z 8 a -8 -4 z 7 a -1 + 5 z 7 a -3 + 17 z 7 a -5 -8 z 7 a -9 + 14 z 6 a -2 + 21 z 6 a -4 + 21 z 6 a -6 + 11 z 6 a -8 -4 z 6 a -10 - z 6 + 11 z 5 a -1 + 6 z 5 a -3 -10 z 5 a -5 + 8 z 5 a -7 + 12 z 5 a -9 - z 5 a -11 -9 z 4 a -2 -13 z 4 a -4 -10 z 4 a -6 -3 z 4 a -8 + 5 z 4 a -10 + 2 z 4 -8 z 3 a -1 -6 z 3 a -3 + 9 z 3 a -5 -6 z 3 a -9 + z 3 a -11 + 2 z 2 a -2 + 5 z 2 a -4 + 5 z 2 a -6 -2 z 2 a -10 - z a -1 -3 z a -3 -4 z a -5 -2 z a -7 -3 a -2 -3 a -4 - a -6 + 2 a -1 z -1 + 3 a -3 z -1 + a -5 z -1$ (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 =

3 is the signature of L11a4. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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