L11a410

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 4 - v u 4 - v 2 w u 4 + 2 v w u 4 - w u 4 -2 v 2 u 3 + 2 v u 3 + v 2 w u 3 -2 v w u 3 + 2 w u 3 - u 3 + 2 v 2 u 2 -2 v u 2 - v 2 w u 2 + 2 v w u 2 -2 w u 2 + u 2 -2 v 2 u + 2 v u + v 2 w u -2 v w u + 2 w u - u + v 2 -2 v + v w - w + 1$ (db)
Jones polynomial	$- q 9 + 3 q 8 -6 q 7 + 9 q 6 -12 q 5 + 13 q 4 -12 q 3 + 12 q 2 -7 q + 6-2 q -1 + q -2$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$z 8 a -4 -2 z 6 a -2 + 6 z 6 a -4 - z 6 a -6 -10 z 4 a -2 + 14 z 4 a -4 -4 z 4 a -6 + z 4 -17 z 2 a -2 + 17 z 2 a -4 -5 z 2 a -6 + 4 z 2 -13 a -2 + 11 a -4 -3 a -6 + 5-5 a -2 z -2 + 4 a -4 z -2 - a -6 z -2 + 2 z -2$ (db)
Kauffman polynomial	$z 10 a -2 + z 10 a -4 + 2 z 9 a -1 + 7 z 9 a -3 + 5 z 9 a -5 + 3 z 8 a -2 + 12 z 8 a -4 + 10 z 8 a -6 + z 8 -9 z 7 a -1 -24 z 7 a -3 -4 z 7 a -5 + 11 z 7 a -7 -33 z 6 a -2 -62 z 6 a -4 -26 z 6 a -6 + 9 z 6 a -8 -6 z 6 + 10 z 5 a -1 + 9 z 5 a -3 -30 z 5 a -5 -23 z 5 a -7 + 6 z 5 a -9 + 67 z 4 a -2 + 87 z 4 a -4 + 19 z 4 a -6 -12 z 4 a -8 + 3 z 4 a -10 + 14 z 4 + 5 z 3 a -1 + 31 z 3 a -3 + 43 z 3 a -5 + 12 z 3 a -7 -4 z 3 a -9 + z 3 a -11 -53 z 2 a -2 -52 z 2 a -4 -12 z 2 a -6 + 3 z 2 a -8 -16 z 2 -13 z a -1 -29 z a -3 -21 z a -5 -4 z a -7 + z a -9 + 21 a -2 + 18 a -4 + 5 a -6 + 9 + 5 a -1 z -1 + 9 a -3 z -1 + 5 a -5 z -1 + a -7 z -1 -5 a -2 z -2 -4 a -4 z -2 - a -6 z -2 -2 z -2$ (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 =

4 is the signature of L11a410. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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