L11n125

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_14,3,15,4 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_4,13,1,14
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 $u$ , $v$ , $w$ , ...)	$u 5 - v u 4 -3 u 4 + 4 u 3 + 4 v u 2 -3 v u - u + v$ (db)
Jones polynomial	$-q^{5/2}+q^{3/2}-\frac{2}{\sqrt{q}}+\frac{4}{q^{3/2}}-\frac{5}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z a 7 + 2 z 3 a 5 + 3 z a 5 + 2 a 5 z -1 - z 5 a 3 -4 z 3 a 3 -8 z a 3 -4 a 3 z -1 + z 5 a + 6 z 3 a + 7 z a + 3 a z -1 - z 3 a -1 -3 z a -1 - a -1 z -1$ (db)
Kauffman polynomial	$- a 5 z 9 - a 3 z 9 -3 a 6 z 8 -4 a 4 z 8 -2 a 2 z 8 - z 8 -3 a 7 z 7 - a 5 z 7 + 2 a 3 z 7 - a z 7 - z 7 a -1 - a 8 z 6 + 10 a 6 z 6 + 18 a 4 z 6 + 14 a 2 z 6 + 7 z 6 + 11 a 7 z 5 + 18 a 5 z 5 + 12 a 3 z 5 + 11 a z 5 + 6 z 5 a -1 + 3 a 8 z 4 -5 a 6 z 4 -23 a 4 z 4 -27 a 2 z 4 -12 z 4 -9 a 7 z 3 -27 a 5 z 3 -34 a 3 z 3 -25 a z 3 -9 z 3 a -1 - a 8 z 2 + a 6 z 2 + 10 a 4 z 2 + 15 a 2 z 2 + 7 z 2 + 3 a 7 z + 15 a 5 z + 23 a 3 z + 16 a z + 5 z a -1 - a 6 -2 a 4 -3 a 2 -1-2 a 5 z -1 -4 a 3 z -1 -3 a z -1 - a -1 z -1$ (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 L11n125. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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