L11a477

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 4 - v 2 w u 4 -3 v 2 u 3 + 2 v u 3 + 3 v 2 w u 3 -2 v w u 3 + 2 v 2 u 2 -4 v u 2 -2 v 2 w u 2 + 4 v w u 2 -2 w u 2 + 2 u 2 + 2 v u -2 v w u + 3 w u -3 u - w + 1$ (db)
Jones polynomial	$- q + 3-5 q -1 + 9 q -2 -10 q -3 + 13 q -4 -12 q -5 + 11 q -6 -8 q -7 + 5 q -8 -2 q -9 + q -10$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$z 4 a 8 + 4 z 2 a 8 + a 8 z -2 + 4 a 8 -2 z 6 a 6 -10 z 4 a 6 -17 z 2 a 6 -2 a 6 z -2 -12 a 6 + z 8 a 4 + 6 z 6 a 4 + 14 z 4 a 4 + 17 z 2 a 4 + a 4 z -2 + 9 a 4 - z 6 a 2 -4 z 4 a 2 -4 z 2 a 2 - a 2$ (db)
Kauffman polynomial	$z 4 a 12 -2 z 2 a 12 + a 12 + 2 z 5 a 11 -2 z 3 a 11 + 3 z 6 a 10 -2 z 4 a 10 + 4 z 7 a 9 -5 z 5 a 9 + 4 z 3 a 9 + 4 z 8 a 8 -6 z 6 a 8 + 6 z 4 a 8 -6 z 2 a 8 - a 8 z -2 + 5 a 8 + 3 z 9 a 7 -3 z 7 a 7 -5 z 5 a 7 + 12 z 3 a 7 -10 z a 7 + 2 a 7 z -1 + z 10 a 6 + 6 z 8 a 6 -29 z 6 a 6 + 43 z 4 a 6 -35 z 2 a 6 -2 a 6 z -2 + 15 a 6 + 6 z 9 a 5 -17 z 7 a 5 + 7 z 5 a 5 + 11 z 3 a 5 -12 z a 5 + 2 a 5 z -1 + z 10 a 4 + 5 z 8 a 4 -33 z 6 a 4 + 51 z 4 a 4 -36 z 2 a 4 - a 4 z -2 + 12 a 4 + 3 z 9 a 3 -9 z 7 a 3 + z 5 a 3 + 9 z 3 a 3 -3 z a 3 + 3 z 8 a 2 -13 z 6 a 2 + 17 z 4 a 2 -9 z 2 a 2 + 2 a 2 + z 7 a -4 z 5 a + 4 z 3 a - z a$ (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 L11a477. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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