L11n115

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_20,8,21,7 X_22,17,5,18 X_18,21,19,22 X_13,17,14,16 X_9,15,10,14 X_15,11,16,10 X_8,20,9,19 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, -7, 8, 11, -2, -6, 7, -8, 6, 4, -5, 9, -3, 5, -4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- u 5 - v u 4 + 4 u 4 + 2 v u 3 -3 u 3 -3 v u 2 + 2 u 2 + 4 v u - u - v$ (db)
Jones polynomial	$q^{7/2}-3 q^{5/2}+5 q^{3/2}-7 \sqrt{q}+\frac{7}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{6}{q^{5/2}}-\frac{5}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z a 5 + 2 a 5 z -1 -3 z 3 a 3 -8 z a 3 -5 a 3 z -1 + 2 z 5 a + 8 z 3 a + 11 z a + 6 a z -1 -3 z 3 a -1 -7 z a -1 -4 a -1 z -1 + z a -3 + a -3 z -1$ (db)
Kauffman polynomial	$- a 3 z 9 - a z 9 -2 a 4 z 8 -6 a 2 z 8 -4 z 8 - a 5 z 7 -2 a 3 z 7 -6 a z 7 -5 z 7 a -1 + 8 a 4 z 6 + 22 a 2 z 6 -2 z 6 a -2 + 12 z 6 + 5 a 5 z 5 + 23 a 3 z 5 + 38 a z 5 + 20 z 5 a -1 -8 a 4 z 4 -18 a 2 z 4 + 5 z 4 a -2 -5 z 4 -9 a 5 z 3 -36 a 3 z 3 -54 a z 3 -30 z 3 a -1 -3 z 3 a -3 + a 4 z 2 + 2 a 2 z 2 -8 z 2 a -2 - z 2 a -4 -6 z 2 + 7 a 5 z + 22 a 3 z + 30 a z + 18 z a -1 + 3 z a -3 + a 4 + a 2 + 3 a -2 + a -4 + 3-2 a 5 z -1 -5 a 3 z -1 -6 a z -1 -4 a -1 z -1 - a -3 z -1$ (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 L11n115. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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