L11a21

(Knotscape image)

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

Visit L11a21's page at the original Knot Atlas.

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 5 + 2 u 5 + 5 v u 4 -5 u 4 -7 v u 3 + 7 u 3 + 7 v u 2 -7 u 2 -5 v u + 5 u + 2 v -2$ (db)
Jones polynomial	$q^{23/2}-4 q^{21/2}+9 q^{19/2}-13 q^{17/2}+17 q^{15/2}-18 q^{13/2}+17 q^{11/2}-15 q^{9/2}+9 q^{7/2}-6 q^{5/2}+2 q^{3/2}-\sqrt{q}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$- z 7 a -5 - z 7 a -7 + z 5 a -3 -4 z 5 a -5 -3 z 5 a -7 + z 5 a -9 + 4 z 3 a -3 -6 z 3 a -5 -2 z 3 a -7 + 2 z 3 a -9 + 5 z a -3 -5 z a -5 - z a -7 + z a -9 + 2 a -3 z -1 -2 a -5 z -1 - a -7 z -1 + a -9 z -1$ (db)
Kauffman polynomial	$- z 10 a -6 - z 10 a -8 -2 z 9 a -5 -7 z 9 a -7 -5 z 9 a -9 -2 z 8 a -4 -5 z 8 a -6 -13 z 8 a -8 -10 z 8 a -10 - z 7 a -3 + 2 z 7 a -5 + 11 z 7 a -7 -4 z 7 a -9 -12 z 7 a -11 + 7 z 6 a -4 + 21 z 6 a -6 + 33 z 6 a -8 + 10 z 6 a -10 -9 z 6 a -12 + 5 z 5 a -3 + 11 z 5 a -5 + 8 z 5 a -7 + 22 z 5 a -9 + 16 z 5 a -11 -4 z 5 a -13 -6 z 4 a -4 -19 z 4 a -6 -19 z 4 a -8 + 4 z 4 a -10 + 9 z 4 a -12 - z 4 a -14 -9 z 3 a -3 -20 z 3 a -5 -15 z 3 a -7 -12 z 3 a -9 -7 z 3 a -11 + z 3 a -13 -2 z 2 a -4 + 4 z 2 a -6 + 6 z 2 a -8 -4 z 2 a -10 -4 z 2 a -12 + 7 z a -3 + 10 z a -5 + 4 z a -7 + 2 z a -9 + z a -11 + 3 a -4 -3 a -8 + a -12 -2 a -3 z -1 -2 a -5 z -1 + a -7 z -1 + a -9 z -1$ (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 =

5 is the signature of L11a21. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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