L11n268

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_3,11,4,10 X_7,17,8,16 X_15,5,16,8 X_11,19,12,18 X_17,9,18,22 X_13,21,14,20 X_19,13,20,12 X_21,15,22,14 X₂₅₃₆ X_9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 4}, {-11, 2, -5, 8, -7, 9, -4, 3, -6, 5, -8, 7, -9, 6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$u 5 + v u 4 -2 v w u 4 + w u 4 -2 u 4 - v u 3 + 2 v w u 3 - w u 3 + 2 u 3 + v u 2 -2 v w u 2 + w u 2 -2 u 2 - v u + 2 v w u - w u + 2 u - v w$ (db)
Jones polynomial	$- q 11 + 2 q 10 -5 q 9 + 6 q 8 -8 q 7 + 10 q 6 -7 q 5 + 7 q 4 -3 q 3 + 3 q 2$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$-2 z 6 a -6 + 3 z 4 a -4 -11 z 4 a -6 + 3 z 4 a -8 + 11 z 2 a -4 -23 z 2 a -6 + 12 z 2 a -8 - z 2 a -10 + 10 a -4 -22 a -6 + 15 a -8 -3 a -10 + 3 a -4 z -2 -8 a -6 z -2 + 7 a -8 z -2 -2 a -10 z -2$ (db)
Kauffman polynomial	$z 9 a -7 + z 9 a -9 + 4 z 8 a -6 + 6 z 8 a -8 + 2 z 8 a -10 + 3 z 7 a -5 + 3 z 7 a -7 + 2 z 7 a -9 + 2 z 7 a -11 -19 z 6 a -6 -24 z 6 a -8 -3 z 6 a -10 + 2 z 6 a -12 -12 z 5 a -5 -23 z 5 a -7 -13 z 5 a -9 - z 5 a -11 + z 5 a -13 + 6 z 4 a -4 + 42 z 4 a -6 + 44 z 4 a -8 + 4 z 4 a -10 -4 z 4 a -12 + 23 z 3 a -5 + 52 z 3 a -7 + 26 z 3 a -9 -6 z 3 a -11 -3 z 3 a -13 -19 z 2 a -4 -46 z 2 a -6 -36 z 2 a -8 -9 z 2 a -10 -24 z a -5 -45 z a -7 -21 z a -9 + 3 z a -11 + 3 z a -13 + 13 a -4 + 28 a -6 + 22 a -8 + 7 a -10 + a -12 + 8 a -5 z -1 + 15 a -7 z -1 + 7 a -9 z -1 - a -11 z -1 - a -13 z -1 -3 a -4 z -2 -8 a -6 z -2 -7 a -8 z -2 -2 a -10 z -2$ (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 L11n268. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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