L11n124

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 3 + 2 u 3 + 5 v u 2 -5 u 2 -5 v u + 5 u + 2 v -2$ (db)
Jones polynomial	$-q^{11/2}+3 q^{9/2}-6 q^{7/2}+8 q^{5/2}-9 q^{3/2}+9 \sqrt{q}-\frac{9}{\sqrt{q}}+\frac{6}{q^{3/2}}-\frac{4}{q^{5/2}}+\frac{1}{q^{7/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-2 z 5 a -1 + 3 a z 3 -7 z 3 a -1 + 3 z 3 a -3 - a 3 z + 5 a z -9 z a -1 + 6 z a -3 - z a -5 + 2 a z -1 -4 a -1 z -1 + 3 a -3 z -1 - a -5 z -1$ (db)
Kauffman polynomial	$-2 z 9 a -1 -2 z 9 a -3 -10 z 8 a -2 -3 z 8 a -4 -7 z 8 -7 a z 7 -5 z 7 a -1 + z 7 a -3 - z 7 a -5 -2 a 2 z 6 + 39 z 6 a -2 + 12 z 6 a -4 + 25 z 6 + 25 a z 5 + 42 z 5 a -1 + 21 z 5 a -3 + 4 z 5 a -5 + a 2 z 4 -42 z 4 a -2 -14 z 4 a -4 -27 z 4 -4 a 3 z 3 -32 a z 3 -56 z 3 a -1 -34 z 3 a -3 -6 z 3 a -5 - a 4 z 2 + 17 z 2 a -2 + 6 z 2 a -4 + 12 z 2 + 3 a 3 z + 15 a z + 26 z a -1 + 18 z a -3 + 4 z a -5 - a 2 -3 a -2 - a -4 -2-2 a z -1 -4 a -1 z -1 -3 a -3 z -1 - a -5 z -1$ (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 L11n124. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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