L11a118

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_14,3,15,4 X_16,10,17,9 X_20,11,21,12 X_8,21,9,22 X_18,7,19,8 X_12,19,13,20 X_10,16,11,15 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 $u$ , $v$ , $w$ , ...)	$- u 5 -3 v u 4 + 7 u 4 + 12 v u 3 -14 u 3 -14 v u 2 + 12 u 2 + 7 v u -3 u - v$ (db)
Jones polynomial	$-\sqrt{q}+\frac{4}{\sqrt{q}}-\frac{10}{q^{3/2}}+\frac{16}{q^{5/2}}-\frac{22}{q^{7/2}}+\frac{24}{q^{9/2}}-\frac{24}{q^{11/2}}+\frac{20}{q^{13/2}}-\frac{15}{q^{15/2}}+\frac{8}{q^{17/2}}-\frac{3}{q^{19/2}}+\frac{1}{q^{21/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- a 11 z -1 + 4 z a 9 + 3 a 9 z -1 -6 z 3 a 7 -8 z a 7 -3 a 7 z -1 + 3 z 5 a 5 + 6 z 3 a 5 + 6 z a 5 + 2 a 5 z -1 + z 5 a 3 -2 z 3 a 3 -4 z a 3 - a 3 z -1 - z 3 a$ (db)
Kauffman polynomial	$- z 6 a 12 + 3 z 4 a 12 -3 z 2 a 12 + a 12 -3 z 7 a 11 + 7 z 5 a 11 -6 z 3 a 11 + 3 z a 11 - a 11 z -1 -5 z 8 a 10 + 8 z 6 a 10 - z 4 a 10 -4 z 2 a 10 + 2 a 10 -5 z 9 a 9 + 2 z 7 a 9 + 13 z 5 a 9 -17 z 3 a 9 + 12 z a 9 -3 a 9 z -1 -2 z 10 a 8 -13 z 8 a 8 + 36 z 6 a 8 -28 z 4 a 8 + 9 z 2 a 8 -13 z 9 a 7 + 15 z 7 a 7 + 13 z 5 a 7 -26 z 3 a 7 + 15 z a 7 -3 a 7 z -1 -2 z 10 a 6 -20 z 8 a 6 + 52 z 6 a 6 -44 z 4 a 6 + 15 z 2 a 6 -2 a 6 -8 z 9 a 5 + z 7 a 5 + 22 z 5 a 5 -26 z 3 a 5 + 11 z a 5 -2 a 5 z -1 -12 z 8 a 4 + 21 z 6 a 4 -16 z 4 a 4 + 5 z 2 a 4 -9 z 7 a 3 + 14 z 5 a 3 -10 z 3 a 3 + 5 z a 3 - a 3 z -1 -4 z 6 a 2 + 4 z 4 a 2 - z 5 a + z 3 a$ (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 =

-3 is the signature of L11a118. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -4$	$i = -2$
$r = -9$	${\mathbb Z}$
$r = -8$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -7$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -6$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -5$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = -4$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{12}$
$r = -3$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = -2$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = -1$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 0$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{7}$
$r = 1$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 2$	${\mathbb Z}_2$	${\mathbb Z}$