L11n402

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_22,16,19,15 X_20,8,21,7 X_8,20,9,19 X_13,18,14,5 X_11,14,12,15 X_17,12,18,13 X_16,22,17,21 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {5, -4, 9, -3}, {10, -1, 4, -5, 11, -2, -7, 8, -6, 7, 3, -9, -8, 6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$u 5 + 2 v u 4 - v w u 4 + w u 4 -2 u 4 -3 v u 3 + 2 v w u 3 -2 w u 3 + 2 u 3 + 2 v u 2 -2 v w u 2 + 3 w u 2 -2 u 2 - v u + 2 v w u -2 w u + u - v w$ (db)
Jones polynomial	$q 3 -2 q 2 + 6 q -8 + 11 q -1 -10 q -2 + 11 q -3 -8 q -4 + 5 q -5 -2 q -6$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$- a 6 z -2 - a 6 - z 4 a 4 + z 2 a 4 + 4 a 4 z -2 + 6 a 4 + z 6 a 2 + 2 z 4 a 2 -2 z 2 a 2 -5 a 2 z -2 -7 a 2 -2 z 4 -4 z 2 + 2 z -2 + z 2 a -2 + 2 a -2$ (db)
Kauffman polynomial	$a 3 z 9 + a z 9 + 4 a 4 z 8 + 7 a 2 z 8 + 3 z 8 + 4 a 5 z 7 + 8 a 3 z 7 + 6 a z 7 + 2 z 7 a -1 + a 6 z 6 -9 a 4 z 6 -19 a 2 z 6 + z 6 a -2 -8 z 6 -9 a 5 z 5 -29 a 3 z 5 -25 a z 5 -5 z 5 a -1 + 4 a 6 z 4 + 19 a 4 z 4 + 27 a 2 z 4 -4 z 4 a -2 + 8 z 4 + 3 a 7 z 3 + 20 a 5 z 3 + 43 a 3 z 3 + 28 a z 3 + 2 z 3 a -1 -4 a 6 z 2 -24 a 4 z 2 -34 a 2 z 2 + 5 z 2 a -2 -9 z 2 -3 a 7 z -17 a 5 z -33 a 3 z -18 a z + z a -1 + 3 a 6 + 16 a 4 + 21 a 2 -2 a -2 + 7 + a 7 z -1 + 5 a 5 z -1 + 9 a 3 z -1 + 5 a z -1 - a 6 z -2 -4 a 4 z -2 -5 a 2 z -2 -2 z -2$ (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 L11n402. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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