L11a474

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 4 - v u 4 - v 2 w u 4 + v w u 4 -2 v 2 u 3 + 2 v u 3 + 2 v 2 w u 3 -2 v w u 3 + w u 3 - u 3 + 2 v 2 u 2 -2 v u 2 -2 v 2 w u 2 + 2 v w u 2 -2 w u 2 + 2 u 2 - v 2 u + 2 v u + v 2 w u -2 v w u + 2 w u -2 u - v + v w - w + 1$ (db)
Jones polynomial	$- q 9 + 3 q 8 -6 q 7 + 9 q 6 -11 q 5 + 13 q 4 -11 q 3 + 11 q 2 -7 q + 5-2 q -1 + q -2$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$z 8 a -4 -2 z 6 a -2 + 6 z 6 a -4 - z 6 a -6 -10 z 4 a -2 + 14 z 4 a -4 -4 z 4 a -6 + z 4 -16 z 2 a -2 + 17 z 2 a -4 -5 z 2 a -6 + 4 z 2 -10 a -2 + 9 a -4 -3 a -6 + 4-2 a -2 z -2 + a -4 z -2 + z -2$ (db)
Kauffman polynomial	$z 10 a -2 + z 10 a -4 + 2 z 9 a -1 + 7 z 9 a -3 + 5 z 9 a -5 + 2 z 8 a -2 + 10 z 8 a -4 + 9 z 8 a -6 + z 8 -10 z 7 a -1 -28 z 7 a -3 -8 z 7 a -5 + 10 z 7 a -7 -29 z 6 a -2 -54 z 6 a -4 -22 z 6 a -6 + 9 z 6 a -8 -6 z 6 + 15 z 5 a -1 + 27 z 5 a -3 -13 z 5 a -5 -19 z 5 a -7 + 6 z 5 a -9 + 61 z 4 a -2 + 76 z 4 a -4 + 12 z 4 a -6 -13 z 4 a -8 + 3 z 4 a -10 + 13 z 4 -5 z 3 a -1 + 3 z 3 a -3 + 19 z 3 a -5 + 6 z 3 a -7 -4 z 3 a -9 + z 3 a -11 -47 z 2 a -2 -44 z 2 a -4 -5 z 2 a -6 + 5 z 2 a -8 -13 z 2 -4 z a -1 -9 z a -3 -6 z a -5 + z a -9 + 15 a -2 + 13 a -4 + 2 a -6 - a -8 + 6 + 2 a -1 z -1 + 2 a -3 z -1 -2 a -2 z -2 - a -4 z -2 - z -2$ (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 L11a474. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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