L11a388

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{3 u v w^2-4 u v w+2 u v+2 u w^3-6 u w^2+6 u w-2 u+2 v w^3-6 v w^2+6 v w-2 v-2 w^3+4 w^2-3 w}{\sqrt{u} \sqrt{v} w^{3/2}}$ (db)
Jones polynomial	$q -2 -3 q -3 + 7 q -4 -10 q -5 + 16 q -6 -15 q -7 + 16 q -8 -13 q -9 + 10 q -10 -6 q -11 + 2 q -12 - q -13$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$- a 14 z -2 + 3 a 12 z -2 + 4 a 12 -6 a 10 z 2 -2 a 10 z -2 -7 a 10 + 3 a 8 z 4 + 3 a 8 z 2 - a 8 z -2 - a 8 + 3 a 6 z 4 + 5 a 6 z 2 + a 6 z -2 + 4 a 6 + a 4 z 4 + a 4 z 2$ (db)
Kauffman polynomial	$a 15 z 7 -5 a 15 z 5 + 9 a 15 z 3 -7 a 15 z + 2 a 15 z -1 + 2 a 14 z 8 -7 a 14 z 6 + 7 a 14 z 4 -2 a 14 z 2 - a 14 z -2 + a 14 + 2 a 13 z 9 - a 13 z 7 -18 a 13 z 5 + 36 a 13 z 3 -27 a 13 z + 8 a 13 z -1 + a 12 z 10 + 5 a 12 z 8 -22 a 12 z 6 + 19 a 12 z 4 -7 a 12 z 2 -3 a 12 z -2 + 5 a 12 + 6 a 11 z 9 -6 a 11 z 7 -27 a 11 z 5 + 49 a 11 z 3 -34 a 11 z + 10 a 11 z -1 + a 10 z 10 + 10 a 10 z 8 -28 a 10 z 6 + 17 a 10 z 4 -5 a 10 z 2 -2 a 10 z -2 + 4 a 10 + 4 a 9 z 9 + 3 a 9 z 7 -23 a 9 z 5 + 23 a 9 z 3 -10 a 9 z + 2 a 9 z -1 + 7 a 8 z 8 -7 a 8 z 6 -4 a 8 z 4 + 9 a 8 z 2 + a 8 z -2 -3 a 8 + 7 a 7 z 7 -6 a 7 z 5 - a 7 z 3 + 4 a 7 z -2 a 7 z -1 + 6 a 6 z 6 -8 a 6 z 4 + 8 a 6 z 2 + a 6 z -2 -4 a 6 + 3 a 5 z 5 -2 a 5 z 3 + a 4 z 4 - a 4 z 2$ (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 =

-4 is the signature of L11a388. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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