L11n338

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_11,18,12,19 X_7,14,8,15 X_13,8,14,9 X_22,20,13,19 X_20,16,21,15 X_16,22,17,21 X_17,12,18,5 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, -3, 9}, {-5, 4, 7, -8, -9, 3, 6, -7, 8, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{(v-1) \left(u v w^3-2 u v w^2+2 u v w+u w^3+v^2+2 v w^2-2 v w+v\right)}{\sqrt{u} v^{3/2} w^{3/2}}$ (db)
Jones polynomial	$- q 2 + 3 q -5 + 6 q -1 -6 q -2 + 6 q -3 -4 q -4 + 3 q -5 + q -6 + q -8$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$2 a 8 z -2 + a 8 -2 z 2 a 6 -5 a 6 z -2 -7 a 6 + 3 z 2 a 4 + 4 a 4 z -2 + 6 a 4 + z 6 a 2 + 4 z 4 a 2 + 5 z 2 a 2 - a 2 z -2 + a 2 - z 4 -2 z 2 -1$ (db)
Kauffman polynomial	$z 8 a 8 -8 z 6 a 8 + 21 z 4 a 8 -24 z 2 a 8 -2 a 8 z -2 + 12 a 8 + z 7 a 7 -10 z 5 a 7 + 25 z 3 a 7 -21 z a 7 + 5 a 7 z -1 + z 8 a 6 -11 z 6 a 6 + 35 z 4 a 6 -41 z 2 a 6 -5 a 6 z -2 + 23 a 6 + 2 z 7 a 5 -15 z 5 a 5 + 42 z 3 a 5 -35 z a 5 + 9 a 5 z -1 + z 8 a 4 -3 z 6 a 4 + 9 z 4 a 4 -12 z 2 a 4 -4 a 4 z -2 + 12 a 4 + 4 z 7 a 3 -11 z 5 a 3 + 17 z 3 a 3 -15 z a 3 + 5 a 3 z -1 + z 8 a 2 + 3 z 6 a 2 -12 z 4 a 2 + 8 z 2 a 2 - a 2 z -2 - a 2 + 3 z 7 a -5 z 5 a -2 z 3 a + a z -1 + 3 z 6 -7 z 4 + 3 z 2 -1 + z 5 a -1 -2 z 3 a -1 + z a -1$ (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 =

-2 is the signature of L11n338. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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