L11n246

(Knotscape image)

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

Planar diagram presentation	X_12,1,13,2 X₃₈₄₉ X_5,14,6,15 X_7,18,8,19 X_9,21,10,20 X_10,11,1,12 X_13,6,14,7 X_17,4,18,5 X_15,11,16,22 X_19,3,20,2 X_21,17,22,16
Gauss code	{1, 10, -2, 8, -3, 7, -4, 2, -5, -6}, {6, -1, -7, 3, -9, 11, -8, 4, -10, 5, -11, 9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 3 + 2 v u 3 - u 3 - v 3 u 2 + 4 v 2 u 2 -6 v u 2 + 2 u 2 + 2 v 3 u -6 v 2 u + 4 v u - u - v 3 + 2 v 2 - v$ (db)
Jones polynomial	$-q^{3/2}+4 \sqrt{q}-\frac{8}{\sqrt{q}}+\frac{10}{q^{3/2}}-\frac{12}{q^{5/2}}+\frac{11}{q^{7/2}}-\frac{10}{q^{9/2}}+\frac{7}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z a 7 + 3 z 3 a 5 + 4 z a 5 + a 5 z -1 -2 z 5 a 3 -6 z 3 a 3 -7 z a 3 - a 3 z -1 + 3 z 3 a + 3 z a - z a -1$ (db)
Kauffman polynomial	$-2 a 5 z 9 -2 a 3 z 9 -5 a 6 z 8 -9 a 4 z 8 -4 a 2 z 8 -4 a 7 z 7 -4 a 5 z 7 -2 a 3 z 7 -2 a z 7 - a 8 z 6 + 12 a 6 z 6 + 23 a 4 z 6 + 10 a 2 z 6 + 11 a 7 z 5 + 24 a 5 z 5 + 15 a 3 z 5 + 2 a z 5 + 2 a 8 z 4 -3 a 6 z 4 -15 a 4 z 4 -14 a 2 z 4 -4 z 4 -7 a 7 z 3 -20 a 5 z 3 -19 a 3 z 3 -7 a z 3 - z 3 a -1 - a 8 z 2 - a 6 z 2 + 2 a 4 z 2 + 4 a 2 z 2 + 2 z 2 + a 7 z + 7 a 5 z + 8 a 3 z + 3 a z + z a -1 + a 4 - a 5 z -1 - a 3 z -1$ (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 =

-1 is the signature of L11n246. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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