L11n123

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_14,3,15,4 X_9,16,10,17 X_20,11,21,12 X_8,21,9,22 X_18,7,19,8 X_12,19,13,20 X_15,10,16,11 X_22,17,5,18 X₂₅₃₆ X_4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 6, -5, -3, 8, 4, -7, 11, -2, -8, 3, 9, -6, 7, -4, 5, -9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- u 5 -3 v u 4 + 3 u 4 + 4 v u 3 -4 u 3 -4 v u 2 + 4 u 2 + 3 v u -3 u - v$ (db)
Jones polynomial	$-\frac{1}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{8}{q^{11/2}}-\frac{11}{q^{13/2}}+\frac{10}{q^{15/2}}-\frac{9}{q^{17/2}}+\frac{7}{q^{19/2}}-\frac{3}{q^{21/2}}+\frac{1}{q^{23/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$- z a 11 -2 a 11 z -1 + 4 z 3 a 9 + 10 z a 9 + 5 a 9 z -1 -3 z 5 a 7 -11 z 3 a 7 -11 z a 7 -3 a 7 z -1 - z 5 a 5 -2 z 3 a 5$ (db)
Kauffman polynomial	$- z 4 a 14 + 2 z 2 a 14 - a 14 -3 z 5 a 13 + 3 z 3 a 13 - z 8 a 12 + z 6 a 12 -6 z 4 a 12 + 6 z 2 a 12 - z 9 a 11 - z 5 a 11 -2 z 3 a 11 + 4 z a 11 -2 a 11 z -1 -5 z 8 a 10 + 10 z 6 a 10 -8 z 4 a 10 -4 z 2 a 10 + 5 a 10 - z 9 a 9 -6 z 7 a 9 + 20 z 5 a 9 -25 z 3 a 9 + 15 z a 9 -5 a 9 z -1 -4 z 8 a 8 + 6 z 6 a 8 + 2 z 4 a 8 -8 z 2 a 8 + 5 a 8 -6 z 7 a 7 + 17 z 5 a 7 -18 z 3 a 7 + 11 z a 7 -3 a 7 z -1 -3 z 6 a 6 + 5 z 4 a 6 - z 5 a 5 + 2 z 3 a 5$ (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 =

-5 is the signature of L11n123. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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