L11n448

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_13,20,14,21 X_16,12,17,11 X_19,12,20,13 X_8,16,5,15 X_14,8,15,7 X_17,19,18,22 X_21,9,22,18 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 7, -6}, {-5, 3, -9, 8}, {11, -2, 4, 5, -3, -7, 6, -4, -8, 9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u + v w u - w u + v x u - v w x u + w x u - x u + u + v - v w + w - v x + v w x - w x + x -1$ (db)
Jones polynomial	$q^{9/2}-3 q^{7/2}+4 q^{5/2}-5 q^{3/2}+4 \sqrt{q}-\frac{7}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{4}{q^{5/2}}-\frac{1}{q^{9/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 5 z -1 + a 5 z -3 -2 z a 3 -4 a 3 z -1 -3 a 3 z -3 + 2 z 3 a + 4 z a + 5 a z -1 + 3 a z -3 - z 5 a -1 -3 z 3 a -1 -3 z a -1 -2 a -1 z -1 - a -1 z -3 + z 3 a -3 + z a -3$ (db)
Kauffman polynomial	$- a z 9 - z 9 a -1 - a 2 z 8 -3 z 8 a -2 -4 z 8 + 3 a z 7 -3 z 7 a -3 + 5 a 2 z 6 + 11 z 6 a -2 - z 6 a -4 + 17 z 6 - a 3 z 5 + a z 5 + 13 z 5 a -1 + 11 z 5 a -3 -9 a 2 z 4 -7 z 4 a -2 + 3 z 4 a -4 -19 z 4 - a 5 z 3 -10 a z 3 -19 z 3 a -1 -8 z 3 a -3 -3 a 4 z 2 - a 2 z 2 - z 2 a -2 - z 2 a -4 + 2 z 2 + 3 a 5 z + 5 a 3 z + 7 a z + 8 z a -1 + 3 z a -3 + 6 a 4 + 11 a 2 + 6-3 a 5 z -1 -6 a 3 z -1 -6 a z -1 -3 a -1 z -1 -3 a 4 z -2 -6 a 2 z -2 -3 z -2 + a 5 z -3 + 3 a 3 z -3 + 3 a z -3 + a -1 z -3$ (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 =

-1 is the signature of L11n448. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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