L11a407

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$4 v 2 u 2 -4 v u 2 -2 v 2 w u 2 + 3 v w u 2 - w u 2 + u 2 -4 v 2 u + 7 v u + 3 v 2 w u -7 v w u + 4 w u -3 u + v 2 -3 v - v 2 w + 4 v w -4 w + 2$ (db)
Jones polynomial	$- q 8 + 4 q 7 -8 q 6 + 13 q 5 -17 q 4 + 19 q 3 -18 q 2 + 16 q -10 + 7 q -1 -2 q -2 + q -3$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 6 a -2 + z 6 a -4 + z 4 a -2 + 2 z 4 a -4 - z 4 a -6 -2 z 4 + a 2 z 2 - z 2 a -2 + 2 z 2 a -4 - z 2 a -6 -4 z 2 + 2 a 2 + a -4 -3 + a 2 z -2 + a -2 z -2 -2 z -2$ (db)
Kauffman polynomial	$z 10 a -2 + z 10 a -4 + 2 z 9 a -1 + 6 z 9 a -3 + 4 z 9 a -5 + 6 z 8 a -2 + 10 z 8 a -4 + 7 z 8 a -6 + 3 z 8 + 2 a z 7 + 5 z 7 a -1 + 4 z 7 a -5 + 7 z 7 a -7 + a 2 z 6 -7 z 6 a -2 -13 z 6 a -4 -7 z 6 a -6 + 4 z 6 a -8 -4 z 6 -4 a z 5 -19 z 5 a -1 -17 z 5 a -3 -14 z 5 a -5 -11 z 5 a -7 + z 5 a -9 -4 a 2 z 4 -8 z 4 a -2 -5 z 4 a -4 -2 z 4 a -6 -6 z 4 a -8 -3 z 4 + 14 z 3 a -1 + 16 z 3 a -3 + 8 z 3 a -5 + 5 z 3 a -7 - z 3 a -9 + 6 a 2 z 2 + 9 z 2 a -2 + 9 z 2 a -4 + 4 z 2 a -6 + 2 z 2 a -8 + 8 z 2 + 4 a z -6 z a -3 -2 z a -5 -4 a 2 -3 a -2 - a -4 - a -6 -6-2 a z -1 -2 a -1 z -1 + a 2 z -2 + a -2 z -2 + 2 z -2$ (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 =

2 is the signature of L11a407. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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