L11a392

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 3 -2 w u 3 + 2 u 3 + 7 v u 2 -4 v w u 2 + 7 w u 2 -5 u 2 -7 v u + 5 v w u -7 w u + 4 u + 2 v -2 v w + 2 w$ (db)
Jones polynomial	$q -2 -4 q -3 + 10 q -4 -13 q -5 + 18 q -6 -18 q -7 + 19 q -8 -14 q -9 + 10 q -10 -6 q -11 + 2 q -12 - q -13$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$- a 14 z -2 + 3 a 12 z -2 + 4 a 12 -6 z 2 a 10 -2 a 10 z -2 -7 a 10 + 3 z 4 a 8 + 2 z 2 a 8 - a 8 z -2 - a 8 + 4 z 4 a 6 + 7 z 2 a 6 + a 6 z -2 + 4 a 6 + z 4 a 4$ (db)
Kauffman polynomial	$z 7 a 15 -5 z 5 a 15 + 9 z 3 a 15 -7 z a 15 + 2 a 15 z -1 + 2 z 8 a 14 -7 z 6 a 14 + 7 z 4 a 14 -2 z 2 a 14 - a 14 z -2 + a 14 + 2 z 9 a 13 - z 7 a 13 -17 z 5 a 13 + 35 z 3 a 13 -27 z a 13 + 8 a 13 z -1 + z 10 a 12 + 6 z 8 a 12 -25 z 6 a 12 + 24 z 4 a 12 -7 z 2 a 12 -3 a 12 z -2 + 5 a 12 + 7 z 9 a 11 -7 z 7 a 11 -27 z 5 a 11 + 50 z 3 a 11 -34 z a 11 + 10 a 11 z -1 + z 10 a 10 + 15 z 8 a 10 -40 z 6 a 10 + 27 z 4 a 10 -8 z 2 a 10 -2 a 10 z -2 + 4 a 10 + 5 z 9 a 9 + 8 z 7 a 9 -35 z 5 a 9 + 26 z 3 a 9 -10 z a 9 + 2 a 9 z -1 + 11 z 8 a 8 -12 z 6 a 8 -2 z 4 a 8 + 4 z 2 a 8 + a 8 z -2 -3 a 8 + 13 z 7 a 7 -16 z 5 a 7 + 2 z 3 a 7 + 4 z a 7 -2 a 7 z -1 + 10 z 6 a 6 -11 z 4 a 6 + 7 z 2 a 6 + a 6 z -2 -4 a 6 + 4 z 5 a 5 + z 4 a 4$ (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 L11a392. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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