L11a375

(Knotscape image)

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

Planar diagram presentation	X_12,1,13,2 X_2,13,3,14 X_14,3,15,4 X_22,7,11,8 X_18,10,19,9 X_20,16,21,15 X_16,6,17,5 X_6,18,7,17 X_10,20,1,19 X_8,11,9,12 X_4,21,5,22
Gauss code	{1, -2, 3, -11, 7, -8, 4, -10, 5, -9}, {10, -1, 2, -3, 6, -7, 8, -5, 9, -6, 11, -4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{u^4 v^4-u^4 v^3-2 u^3 v^4+6 u^3 v^3-5 u^3 v^2+u^3 v+u^2 v^4-6 u^2 v^3+11 u^2 v^2-6 u^2 v+u^2+u v^3-5 u v^2+6 u v-2 u-v+1}{u^2 v^2}$ (db)
Jones polynomial	$q^{9/2}-3 q^{7/2}+7 q^{5/2}-12 q^{3/2}+15 \sqrt{q}-\frac{19}{\sqrt{q}}+\frac{18}{q^{3/2}}-\frac{16}{q^{5/2}}+\frac{12}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- a z 9 + a 3 z 7 -7 a z 7 + z 7 a -1 + 5 a 3 z 5 -19 a z 5 + 5 z 5 a -1 + 9 a 3 z 3 -24 a z 3 + 9 z 3 a -1 + 6 a 3 z -13 a z + 6 z a -1 + a 3 z -1 - a z -1$ (db)
Kauffman polynomial	$-2 a 2 z 10 -2 z 10 -5 a 3 z 9 -10 a z 9 -5 z 9 a -1 -6 a 4 z 8 -5 a 2 z 8 -5 z 8 a -2 -4 z 8 -5 a 5 z 7 + 7 a 3 z 7 + 27 a z 7 + 12 z 7 a -1 -3 z 7 a -3 -3 a 6 z 6 + 9 a 4 z 6 + 17 a 2 z 6 + 13 z 6 a -2 - z 6 a -4 + 19 z 6 - a 7 z 5 + 7 a 5 z 5 -9 a 3 z 5 -38 a z 5 -13 z 5 a -1 + 8 z 5 a -3 + 5 a 6 z 4 -7 a 4 z 4 -21 a 2 z 4 -10 z 4 a -2 + 3 z 4 a -4 -22 z 4 + 2 a 7 z 3 -2 a 5 z 3 + 10 a 3 z 3 + 32 a z 3 + 13 z 3 a -1 -5 z 3 a -3 -2 a 6 z 2 + 4 a 4 z 2 + 10 a 2 z 2 + 4 z 2 a -2 -2 z 2 a -4 + 10 z 2 - a 7 z -6 a 3 z -14 a z -6 z a -1 + z a -3 - a 2 + a 3 z -1 + a z -1$ (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 L11a375. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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