L10n45

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Planar diagram presentation	X₈₁₉₂ X_18,9,19,10 X₆₇₁₈ X_20,14,7,13 X_5,13,6,12 X_3,10,4,11 X_15,5,16,4 X_11,16,12,17 X_14,20,15,19 X_17,2,18,3
Gauss code	{1, 10, -6, 7, -5, -3}, {3, -1, 2, 6, -8, 5, 4, -9, -7, 8, -10, -2, 9, -4}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(2)^{2}t(1)^{2}-t(2)t(1)^{2}-t(2)t(1)-t(2)+1}{t(1)t(2)}}$ (db)
Jones polynomial	$-q^{5/2}+q^{3/2}-{\sqrt {q}}+{\frac {1}{\sqrt {q}}}-{\frac {1}{q^{3/2}}}-{\frac {1}{q^{7/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{5}z^{-1}-a^{3}z^{3}-4a^{3}z-2a^{3}z^{-1}+az^{5}+5az^{3}-z^{3}a^{-1}+6az+2az^{-1}-3za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{5}z+a^{5}z^{-1}-a^{4}z^{2}+a^{3}z^{7}-6a^{3}z^{5}+10a^{3}z^{3}-7a^{3}z+2a^{3}z^{-1}+a^{2}z^{8}-6a^{2}z^{6}+10a^{2}z^{4}-6a^{2}z^{2}+a^{2}+2az^{7}+z^{7}a^{-1}-12az^{5}-6z^{5}a^{-1}+20az^{3}+10z^{3}a^{-1}-11az-5za^{-1}+2az^{-1}+a^{-1}z^{-1}+z^{8}-6z^{6}+10z^{4}-5z^{2}$ (db)

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums

\chi

(fixed

j

, alternation over

r

).

-2

-1

0

1

2

3

4

χ

6

1

4

0

2

1

0

1

0

-2

1

0

-4

1

-6

1

-8

1

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