L11a386

(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_18,12,19,11 X_22,20,9,19 X_20,16,21,15 X_16,22,17,21 X_12,18,13,17 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -4}, {11, -2, 5, -9, 4, -3, 7, -8, 9, -5, 6, -7, 8, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{u v w^3-4 u v w^2+5 u v w-3 u v-3 u w^3+7 u w^2-7 u w+3 u-3 v w^3+7 v w^2-7 v w+3 v+3 w^3-5 w^2+4 w-1}{\sqrt{u} \sqrt{v} w^{3/2}}$ (db)
Jones polynomial	$- q 5 + 4 q 4 -10 q 3 + 15 q 2 -20 q + 22-20 q -1 + 19 q -2 -11 q -3 + 7 q -4 -2 q -5 + q -6$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a 6 z -2 + a 6 -3 z 2 a 4 - a 4 z -2 -4 a 4 + 3 z 4 a 2 + 3 z 2 a 2 -2 a 2 z -2 - a 2 - z 6 + 4 z 2 + 3 z -2 + 7 + 2 z 4 a -2 - a -2 z -2 -3 a -2 - z 2 a -4$ (db)
Kauffman polynomial	$a 2 z 10 + z 10 + 3 a 3 z 9 + 8 a z 9 + 5 z 9 a -1 + 3 a 4 z 8 + 12 a 2 z 8 + 10 z 8 a -2 + 19 z 8 + 2 a 5 z 7 + a 3 z 7 + 2 a z 7 + 12 z 7 a -1 + 9 z 7 a -3 + a 6 z 6 -3 a 4 z 6 -30 a 2 z 6 -13 z 6 a -2 + 4 z 6 a -4 -43 z 6 -4 a 5 z 5 -12 a 3 z 5 -39 a z 5 -47 z 5 a -1 -15 z 5 a -3 + z 5 a -5 -4 a 6 z 4 -4 a 4 z 4 + 26 a 2 z 4 + 4 z 4 a -2 -4 z 4 a -4 + 34 z 4 + 16 a 3 z 3 + 53 a z 3 + 50 z 3 a -1 + 12 z 3 a -3 - z 3 a -5 + 6 a 6 z 2 + 6 a 4 z 2 -11 a 2 z 2 -2 z 2 a -2 + z 2 a -4 -14 z 2 + 4 a 5 z -10 a 3 z -34 a z -27 z a -1 -7 z a -3 -4 a 6 -3 a 4 + 4 a 2 + a -2 + 5-2 a 5 z -1 + 2 a 3 z -1 + 10 a z -1 + 8 a -1 z -1 + 2 a -3 z -1 + a 6 z -2 + a 4 z -2 -2 a 2 z -2 - a -2 z -2 -3 z -2$ (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 =

0 is the signature of L11a386. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -1$	$i = 1$
$r = -6$	${\mathbb Z}$
$r = -5$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2$	${\mathbb Z}^{4}$
$r = -3$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = -1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 0$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{10}$
$r = 1$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 2$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 4$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 5$	${\mathbb Z}_2$	${\mathbb Z}$