L10a49

(Knotscape image)

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

Visit L10a49's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_10,5,11,6 X_14,3,15,4 X_18,11,19,12 X_20,13,5,14 X_12,19,13,20 X_2,9,3,10 X_8,15,9,16
Gauss code	{1, -9, 5, -3}, {4, -1, 2, -10, 9, -4, 6, -8, 7, -5, 10, -2, 3, -6, 8, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- u 5 -4 v u 4 + 4 u 4 + 6 v u 3 -6 u 3 -6 v u 2 + 6 u 2 + 4 v u -4 u - v$ (db)
Jones polynomial	$-\frac{1}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{9}{q^{11/2}}-\frac{14}{q^{13/2}}+\frac{14}{q^{15/2}}-\frac{13}{q^{17/2}}+\frac{11}{q^{19/2}}-\frac{7}{q^{21/2}}+\frac{4}{q^{23/2}}-\frac{1}{q^{25/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$z 3 a 11 -2 a 11 z -1 - z 5 a 9 + z 3 a 9 + 8 z a 9 + 5 a 9 z -1 -3 z 5 a 7 -10 z 3 a 7 -10 z a 7 -3 a 7 z -1 - z 5 a 5 -2 z 3 a 5$ (db)
Kauffman polynomial	$- z 5 a 15 + z 3 a 15 -4 z 6 a 14 + 7 z 4 a 14 -2 z 2 a 14 - a 14 -6 z 7 a 13 + 10 z 5 a 13 -3 z 3 a 13 + z a 13 -5 z 8 a 12 + 5 z 6 a 12 + z 4 a 12 + z 2 a 12 -2 z 9 a 11 -6 z 7 a 11 + 16 z 5 a 11 -11 z 3 a 11 + 5 z a 11 -2 a 11 z -1 -10 z 8 a 10 + 20 z 6 a 10 -13 z 4 a 10 -4 z 2 a 10 + 5 a 10 -2 z 9 a 9 -6 z 7 a 9 + 22 z 5 a 9 -27 z 3 a 9 + 14 z a 9 -5 a 9 z -1 -5 z 8 a 8 + 8 z 6 a 8 -2 z 4 a 8 -7 z 2 a 8 + 5 a 8 -6 z 7 a 7 + 16 z 5 a 7 -18 z 3 a 7 + 10 z a 7 -3 a 7 z -1 -3 z 6 a 6 + 5 z 4 a 6 - z 5 a 5 + 2 z 3 a 5$ (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 L10a49. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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