L11a235

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_12,3,13,4 X_20,13,21,14 X_16,9,17,10 X_10,19,11,20 X_22,15,7,16 X_14,21,15,22 X_18,5,19,6 X₂₇₃₈ X_4,11,5,12 X_6,17,1,18
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 4, -5, 10, -2, 3, -7, 6, -4, 11, -8, 5, -3, 7, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{-3 u^2 v^4+4 u^2 v^3-4 u^2 v^2+3 u^2 v-u^2-3 u v^5+6 u v^4-7 u v^3+6 u v^2-3 u v-v^6+3 v^5-4 v^4+4 v^3-3 v^2}{u v^3}$ (db)
Jones polynomial	$-\frac{1}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{11}{q^{11/2}}-\frac{16}{q^{13/2}}+\frac{17}{q^{15/2}}-\frac{18}{q^{17/2}}+\frac{15}{q^{19/2}}-\frac{11}{q^{21/2}}+\frac{7}{q^{23/2}}-\frac{3}{q^{25/2}}+\frac{1}{q^{27/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$- z a 13 -2 a 13 z -1 + 4 z 3 a 11 + 10 z a 11 + 5 a 11 z -1 -3 z 5 a 9 -9 z 3 a 9 -8 z a 9 -3 a 9 z -1 -3 z 5 a 7 -8 z 3 a 7 -4 z a 7 - z 5 a 5 -2 z 3 a 5$ (db)
Kauffman polynomial	$- z 6 a 16 + 3 z 4 a 16 -3 z 2 a 16 + a 16 -3 z 7 a 15 + 8 z 5 a 15 -5 z 3 a 15 -4 z 8 a 14 + 7 z 6 a 14 + 2 z 4 a 14 -4 z 2 a 14 -4 z 9 a 13 + 7 z 7 a 13 -5 z 5 a 13 + 10 z 3 a 13 -7 z a 13 + 2 a 13 z -1 -2 z 10 a 12 -2 z 8 a 12 + 12 z 6 a 12 -17 z 4 a 12 + 17 z 2 a 12 -5 a 12 -10 z 9 a 11 + 31 z 7 a 11 -51 z 5 a 11 + 47 z 3 a 11 -22 z a 11 + 5 a 11 z -1 -2 z 10 a 10 -5 z 8 a 10 + 23 z 6 a 10 -36 z 4 a 10 + 21 z 2 a 10 -5 a 10 -6 z 9 a 9 + 15 z 7 a 9 -23 z 5 a 9 + 18 z 3 a 9 -11 z a 9 + 3 a 9 z -1 -7 z 8 a 8 + 16 z 6 a 8 -15 z 4 a 8 + 3 z 2 a 8 -6 z 7 a 7 + 14 z 5 a 7 -12 z 3 a 7 + 4 z a 7 -3 z 6 a 6 + 5 z 4 a 6 - z 5 a 5 + 2 z 3 a 5$ (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 =

-5 is the signature of L11a235. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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