L10a38

(Knotscape image)

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

Visit L10a38's page at the original Knot Atlas.

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- u 5 -3 v u 4 + 4 u 4 + 6 v u 3 -7 u 3 -7 v u 2 + 6 u 2 + 4 v u -3 u - v$ (db)
Jones polynomial	$-\frac{1}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{10}{q^{11/2}}-\frac{14}{q^{13/2}}+\frac{14}{q^{15/2}}-\frac{13}{q^{17/2}}+\frac{11}{q^{19/2}}-\frac{6}{q^{21/2}}+\frac{3}{q^{23/2}}-\frac{1}{q^{25/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a 13 z -1 -4 z a 11 -5 a 11 z -1 + 6 z 3 a 9 + 14 z a 9 + 8 a 9 z -1 -3 z 5 a 7 -10 z 3 a 7 -11 z a 7 -4 a 7 z -1 - z 5 a 5 -2 z 3 a 5 - z a 5$ (db)
Kauffman polynomial	$- z 5 a 15 + 2 z 3 a 15 - z a 15 -3 z 6 a 14 + 6 z 4 a 14 -5 z 2 a 14 + 2 a 14 -4 z 7 a 13 + 4 z 5 a 13 + z 3 a 13 - z a 13 - a 13 z -1 -3 z 8 a 12 -5 z 6 a 12 + 21 z 4 a 12 -22 z 2 a 12 + 9 a 12 - z 9 a 11 -11 z 7 a 11 + 23 z 5 a 11 -16 z 3 a 11 + 9 z a 11 -5 a 11 z -1 -7 z 8 a 10 - z 6 a 10 + 29 z 4 a 10 -35 z 2 a 10 + 14 a 10 - z 9 a 9 -13 z 7 a 9 + 32 z 5 a 9 -32 z 3 a 9 + 22 z a 9 -8 a 9 z -1 -4 z 8 a 8 -2 z 6 a 8 + 18 z 4 a 8 -19 z 2 a 8 + 8 a 8 -6 z 7 a 7 + 13 z 5 a 7 -15 z 3 a 7 + 12 z a 7 -4 a 7 z -1 -3 z 6 a 6 + 4 z 4 a 6 - z 2 a 6 - z 5 a 5 + 2 z 3 a 5 - z a 5$ (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 L10a38. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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