L11a85

(Knotscape image)

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

Visit L11a85's page at the original Knot Atlas.

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + 2 u 5 + 4 v u 4 -4 u 4 -5 v u 3 + 5 u 3 + 5 v u 2 -5 u 2 -4 v u + 4 u + 2 v -1$ (db)
Jones polynomial	$-\frac{1}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{6}{q^{5/2}}+\frac{9}{q^{7/2}}-\frac{12}{q^{9/2}}+\frac{12}{q^{11/2}}-\frac{14}{q^{13/2}}+\frac{11}{q^{15/2}}-\frac{8}{q^{17/2}}+\frac{5}{q^{19/2}}-\frac{2}{q^{21/2}}+\frac{1}{q^{23/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$- z a 11 -2 a 11 z -1 + 3 z 3 a 9 + 9 z a 9 + 5 a 9 z -1 -3 z 5 a 7 -11 z 3 a 7 -11 z a 7 -3 a 7 z -1 + z 7 a 5 + 4 z 5 a 5 + 5 z 3 a 5 + 3 z a 5 - z 5 a 3 -3 z 3 a 3 -2 z a 3$ (db)
Kauffman polynomial	$- z 4 a 14 + 2 z 2 a 14 - a 14 -2 z 5 a 13 + 2 z 3 a 13 -3 z 6 a 12 + 2 z 4 a 12 -4 z 7 a 11 + 5 z 5 a 11 -6 z 3 a 11 + 5 z a 11 -2 a 11 z -1 -4 z 8 a 10 + 4 z 6 a 10 + 3 z 4 a 10 -10 z 2 a 10 + 5 a 10 -3 z 9 a 9 + z 7 a 9 + 12 z 5 a 9 -20 z 3 a 9 + 14 z a 9 -5 a 9 z -1 - z 10 a 8 -6 z 8 a 8 + 22 z 6 a 8 -15 z 4 a 8 -3 z 2 a 8 + 5 a 8 -6 z 9 a 7 + 14 z 7 a 7 + z 5 a 7 -13 z 3 a 7 + 8 z a 7 -3 a 7 z -1 - z 10 a 6 -5 z 8 a 6 + 27 z 6 a 6 -29 z 4 a 6 + 10 z 2 a 6 -3 z 9 a 5 + 8 z 7 a 5 -6 z 3 a 5 + z a 5 -3 z 8 a 4 + 12 z 6 a 4 -14 z 4 a 4 + 5 z 2 a 4 - z 7 a 3 + 4 z 5 a 3 -5 z 3 a 3 + 2 z a 3$ (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 L11a85. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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