L10a77

(Knotscape image)

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

Visit L10a77's page at the original Knot Atlas.

Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_16,5,17,6 X_14,7,15,8 X_18,11,19,12 X_20,15,7,16 X_12,17,13,18 X_4,13,5,14 X_6,19,1,20
Gauss code	{1, -2, 3, -9, 4, -10}, {5, -1, 2, -3, 6, -8, 9, -5, 7, -4, 8, -6, 10, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 4 -4 v 2 u 3 + 5 v u 3 - u 3 + 4 v 2 u 2 -9 v u 2 + 4 u 2 - v 2 u + 5 v u -4 u - v$ (db)
Jones polynomial	$-\frac{1}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{9}{q^{11/2}}-\frac{13}{q^{13/2}}+\frac{13}{q^{15/2}}-\frac{12}{q^{17/2}}+\frac{10}{q^{19/2}}-\frac{6}{q^{21/2}}+\frac{3}{q^{23/2}}-\frac{1}{q^{25/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$z 3 a 11 + z a 11 - a 11 z -1 - z 5 a 9 + 5 z a 9 + 3 a 9 z -1 -3 z 5 a 7 -10 z 3 a 7 -9 z a 7 -2 a 7 z -1 - z 5 a 5 -2 z 3 a 5$ (db)
Kauffman polynomial	$- z 5 a 15 + 2 z 3 a 15 - z a 15 -3 z 6 a 14 + 6 z 4 a 14 -3 z 2 a 14 -4 z 7 a 13 + 5 z 5 a 13 + z 3 a 13 - z a 13 -4 z 8 a 12 + 5 z 6 a 12 -3 z 4 a 12 + 3 z 2 a 12 + a 12 -2 z 9 a 11 -2 z 7 a 11 + 5 z 5 a 11 - z 3 a 11 - a 11 z -1 -9 z 8 a 10 + 20 z 6 a 10 -18 z 4 a 10 + 2 z 2 a 10 + 3 a 10 -2 z 9 a 9 -4 z 7 a 9 + 16 z 5 a 9 -19 z 3 a 9 + 9 z a 9 -3 a 9 z -1 -5 z 8 a 8 + 9 z 6 a 8 -4 z 4 a 8 -4 z 2 a 8 + 3 a 8 -6 z 7 a 7 + 16 z 5 a 7 -17 z 3 a 7 + 9 z a 7 -2 a 7 z -1 -3 z 6 a 6 + 5 z 4 a 6 - z 5 a 5 + 2 z 3 a 5$ (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 L10a77. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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