L11a81

(Knotscape image)

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

Visit L11a81's page at the original Knot Atlas.

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 u 5 -5 v u 4 + 7 u 4 + 11 v u 3 -12 u 3 -12 v u 2 + 11 u 2 + 7 v u -5 u -2 v$ (db)
Jones polynomial	$-\frac{1}{q^{5/2}}+\frac{4}{q^{7/2}}-\frac{11}{q^{9/2}}+\frac{16}{q^{11/2}}-\frac{22}{q^{13/2}}+\frac{24}{q^{15/2}}-\frac{24}{q^{17/2}}+\frac{20}{q^{19/2}}-\frac{14}{q^{21/2}}+\frac{8}{q^{23/2}}-\frac{3}{q^{25/2}}+\frac{1}{q^{27/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$- z a 13 - a 13 z -1 + 3 z 3 a 11 + 4 z a 11 + a 11 z -1 -2 z 5 a 9 - z 3 a 9 + 4 z a 9 + 2 a 9 z -1 -4 z 5 a 7 -11 z 3 a 7 -9 z a 7 -2 a 7 z -1 - z 5 a 5 - z 3 a 5$ (db)
Kauffman polynomial	$- z 6 a 16 + 3 z 4 a 16 -3 z 2 a 16 + a 16 -3 z 7 a 15 + 7 z 5 a 15 -5 z 3 a 15 + z a 15 -5 z 8 a 14 + 9 z 6 a 14 -4 z 4 a 14 -5 z 9 a 13 + 4 z 7 a 13 + 5 z 5 a 13 -4 z 3 a 13 - z a 13 + a 13 z -1 -2 z 10 a 12 -12 z 8 a 12 + 34 z 6 a 12 -31 z 4 a 12 + 15 z 2 a 12 -3 a 12 -13 z 9 a 11 + 18 z 7 a 11 -3 z 5 a 11 + 2 z 3 a 11 -4 z a 11 + a 11 z -1 -2 z 10 a 10 -19 z 8 a 10 + 45 z 6 a 10 -31 z 4 a 10 + 7 z 2 a 10 -8 z 9 a 9 + z 7 a 9 + 17 z 5 a 9 -15 z 3 a 9 + 7 z a 9 -2 a 9 z -1 -12 z 8 a 8 + 17 z 6 a 8 -4 z 4 a 8 -5 z 2 a 8 + 3 a 8 -10 z 7 a 7 + 17 z 5 a 7 -15 z 3 a 7 + 9 z a 7 -2 a 7 z -1 -4 z 6 a 6 + 3 z 4 a 6 - z 5 a 5 + z 3 a 5$ (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 L11a81. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -6$	$i = -4$
$r = -11$	${\mathbb Z}$
$r = -10$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -9$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -8$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -7$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = -6$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = -5$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = -4$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{13}$
$r = -3$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = -2$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = -1$	${\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 0$	${\mathbb Z}$	${\mathbb Z}$