L11a488

(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_16,10,17,9 X_22,18,19,17 X_20,14,21,13 X_14,20,15,19 X_18,22,5,21 X_8,16,9,15 X₂₅₃₆ X_4,12,1,11
Gauss code	{1, -10, 2, -11}, {7, -6, 8, -5}, {10, -1, 3, -9, 4, -2, 11, -3, 6, -7, 9, -4, 5, -8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + u 5 + 3 v u 4 -2 v w u 4 + 2 w u 4 -3 u 4 -5 v u 3 + 5 v w u 3 -5 w u 3 + 5 u 3 + 5 v u 2 -5 v w u 2 + 5 w u 2 -5 u 2 -2 v u + 3 v w u -3 w u + 2 u - v w + w$ (db)
Jones polynomial	$- q 11 + 3 q 10 -8 q 9 + 14 q 8 -18 q 7 + 21 q 6 -20 q 5 + 19 q 4 -12 q 3 + 8 q 2 -3 q + 1$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$- z 6 a -4 -2 z 6 a -6 + z 4 a -2 - z 4 a -4 -7 z 4 a -6 + 3 z 4 a -8 + 2 z 2 a -2 + 3 z 2 a -4 -12 z 2 a -6 + 8 z 2 a -8 - z 2 a -10 + a -2 + 5 a -4 -13 a -6 + 9 a -8 -2 a -10 + 2 a -4 z -2 -5 a -6 z -2 + 4 a -8 z -2 - a -10 z -2$ (db)
Kauffman polynomial	$z 10 a -6 + z 10 a -8 + 4 z 9 a -5 + 9 z 9 a -7 + 5 z 9 a -9 + 5 z 8 a -4 + 16 z 8 a -6 + 19 z 8 a -8 + 8 z 8 a -10 + 3 z 7 a -3 -5 z 7 a -7 + 4 z 7 a -9 + 6 z 7 a -11 + z 6 a -2 -11 z 6 a -4 -51 z 6 a -6 -56 z 6 a -8 -14 z 6 a -10 + 3 z 6 a -12 -7 z 5 a -3 -18 z 5 a -5 -29 z 5 a -7 -28 z 5 a -9 -9 z 5 a -11 + z 5 a -13 -3 z 4 a -2 + 10 z 4 a -4 + 64 z 4 a -6 + 71 z 4 a -8 + 16 z 4 a -10 -4 z 4 a -12 + 4 z 3 a -3 + 24 z 3 a -5 + 50 z 3 a -7 + 39 z 3 a -9 + 7 z 3 a -11 -2 z 3 a -13 + 3 z 2 a -2 -11 z 2 a -4 -48 z 2 a -6 -48 z 2 a -8 -13 z 2 a -10 + z 2 a -12 -17 z a -5 -35 z a -7 -23 z a -9 -4 z a -11 + z a -13 - a -2 + 8 a -4 + 22 a -6 + 19 a -8 + 5 a -10 + 5 a -5 z -1 + 9 a -7 z -1 + 5 a -9 z -1 + a -11 z -1 -2 a -4 z -2 -5 a -6 z -2 -4 a -8 z -2 - a -10 z -2$ (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 L11a488. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 3$	$i = 5$
$r = -2$	${\mathbb Z}$
$r = -1$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = 0$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{4}$
$r = 1$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 2$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 3$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 4$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{11}$
$r = 5$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 6$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 7$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 8$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 9$	${\mathbb Z}_2$	${\mathbb Z}$