L11n410

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_3,13,4,12 X_13,22,14,19 X_7,20,8,21 X_19,10,20,11 X_9,16,10,17 X_17,14,18,15 X_15,8,16,9 X_21,18,22,5 X₂₅₃₆ X_11,1,12,4
Gauss code	{1, -10, -2, 11}, {-5, 4, -9, 3}, {10, -1, -4, 8, -6, 5, -11, 2, -3, 7, -8, 6, -7, 9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$2 v u 3 -2 v w u 3 + w u 3 - u 3 -4 v u 2 + 4 v w u 2 -3 w u 2 + 3 u 2 + 3 v u -3 v w u + 4 w u -4 u - v + v w -2 w + 2$ (db)
Jones polynomial	$1-3 q -1 + 7 q -2 -9 q -3 + 14 q -4 -13 q -5 + 13 q -6 -10 q -7 + 7 q -8 -3 q -9$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$- a 10 z -2 - a 10 + z 4 a 8 + 3 z 2 a 8 + 4 a 8 z -2 + 6 a 8 - z 6 a 6 -3 z 4 a 6 -6 z 2 a 6 -5 a 6 z -2 -10 a 6 - z 6 a 4 -2 z 4 a 4 + z 2 a 4 + 2 a 4 z -2 + 4 a 4 + z 4 a 2 + 2 z 2 a 2 + a 2$ (db)
Kauffman polynomial	$6 z 3 a 11 -5 z a 11 + a 11 z -1 + 3 z 6 a 10 + 3 z 4 a 10 -6 z 2 a 10 - a 10 z -2 + 3 a 10 + 8 z 7 a 9 -19 z 5 a 9 + 29 z 3 a 9 -20 z a 9 + 5 a 9 z -1 + 7 z 8 a 8 -16 z 6 a 8 + 25 z 4 a 8 -23 z 2 a 8 -4 a 8 z -2 + 13 a 8 + 2 z 9 a 7 + 9 z 7 a 7 -33 z 5 a 7 + 41 z 3 a 7 -30 z a 7 + 9 a 7 z -1 + 11 z 8 a 6 -29 z 6 a 6 + 30 z 4 a 6 -25 z 2 a 6 -5 a 6 z -2 + 16 a 6 + 2 z 9 a 5 + 4 z 7 a 5 -22 z 5 a 5 + 23 z 3 a 5 -15 z a 5 + 5 a 5 z -1 + 4 z 8 a 4 -9 z 6 a 4 + 5 z 4 a 4 -5 z 2 a 4 -2 a 4 z -2 + 6 a 4 + 3 z 7 a 3 -8 z 5 a 3 + 5 z 3 a 3 + z 6 a 2 -3 z 4 a 2 + 3 z 2 a 2 - a 2$ (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 L11n410. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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