L11a413

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + v u 4 - v w u 4 + 3 v 2 u 3 -4 v u 3 -2 v 2 w u 3 + 4 v w u 3 - w u 3 + u 3 -3 v 2 u 2 + 6 v u 2 + 2 v 2 w u 2 -6 v w u 2 + 3 w u 2 -2 u 2 + v 2 u -4 v u - v 2 w u + 4 v w u -3 w u + 2 u + v - v w + w$ (db)
Jones polynomial	$q -3 -3 q -4 + 8 q -5 -11 q -6 + 17 q -7 -18 q -8 + 19 q -9 -16 q -10 + 12 q -11 -7 q -12 + 3 q -13 - q -14$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$- a 14 z -2 - a 14 + 4 z 2 a 12 + 4 a 12 z -2 + 9 a 12 -6 z 4 a 10 -20 z 2 a 10 -5 a 10 z -2 -19 a 10 + 3 z 6 a 8 + 13 z 4 a 8 + 19 z 2 a 8 + 2 a 8 z -2 + 11 a 8 + z 6 a 6 + 3 z 4 a 6 + 2 z 2 a 6$ (db)
Kauffman polynomial	$z 5 a 17 -2 z 3 a 17 + z a 17 + 3 z 6 a 16 -5 z 4 a 16 + 3 z 2 a 16 - a 16 + 5 z 7 a 15 -7 z 5 a 15 + 4 z 3 a 15 -2 z a 15 + a 15 z -1 + 5 z 8 a 14 -2 z 6 a 14 -6 z 4 a 14 + 5 z 2 a 14 - a 14 z -2 + 3 z 9 a 13 + 8 z 7 a 13 -27 z 5 a 13 + 31 z 3 a 13 -19 z a 13 + 5 a 13 z -1 + z 10 a 12 + 11 z 8 a 12 -24 z 6 a 12 + 21 z 4 a 12 -16 z 2 a 12 -4 a 12 z -2 + 13 a 12 + 7 z 9 a 11 -2 z 7 a 11 -29 z 5 a 11 + 50 z 3 a 11 -35 z a 11 + 9 a 11 z -1 + z 10 a 10 + 12 z 8 a 10 -39 z 6 a 10 + 51 z 4 a 10 -43 z 2 a 10 -5 a 10 z -2 + 22 a 10 + 4 z 9 a 9 -2 z 7 a 9 -17 z 5 a 9 + 28 z 3 a 9 -19 z a 9 + 5 a 9 z -1 + 6 z 8 a 8 -19 z 6 a 8 + 26 z 4 a 8 -23 z 2 a 8 -2 a 8 z -2 + 11 a 8 + 3 z 7 a 7 -7 z 5 a 7 + 3 z 3 a 7 + z 6 a 6 -3 z 4 a 6 + 2 z 2 a 6$ (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 =

-6 is the signature of L11a413. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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