L11n20

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_17,1,18,4 X_5,12,6,13 X₃₈₄₉ X_13,22,14,5 X_21,14,22,15 X_9,18,10,19 X_11,20,12,21 X_19,10,20,11 X_2,16,3,15
Gauss code	{1, -11, -5, 3}, {-4, -1, 2, 5, -8, 10, -9, 4, -6, 7, 11, -2, -3, 8, -10, 9, -7, 6}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(2t(2)^{2}-t(2)+2\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {2}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {6}{q^{7/2}}}+{\frac {6}{q^{9/2}}}-{\frac {7}{q^{11/2}}}+{\frac {6}{q^{13/2}}}-{\frac {5}{q^{15/2}}}+{\frac {3}{q^{17/2}}}-{\frac {1}{q^{19/2}}}+{\frac {1}{q^{21/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}\left(-z^{3}\right)-3a^{9}z-2a^{9}z^{-1}+a^{7}z^{5}+4a^{7}z^{3}+7a^{7}z+4a^{7}z^{-1}+a^{5}z^{5}+2a^{5}z^{3}-a^{5}z^{-1}-2a^{3}z^{3}-4a^{3}z-a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{12}z^{6}-5a^{12}z^{4}+7a^{12}z^{2}-2a^{12}+a^{11}z^{7}-3a^{11}z^{5}+a^{11}z^{3}+a^{11}z+a^{10}z^{8}-2a^{10}z^{6}-a^{10}z^{4}+2a^{10}z^{2}-a^{10}+a^{9}z^{9}-4a^{9}z^{7}+11a^{9}z^{5}-17a^{9}z^{3}+8a^{9}z-2a^{9}z^{-1}+3a^{8}z^{8}-11a^{8}z^{6}+21a^{8}z^{4}-21a^{8}z^{2}+6a^{8}+a^{7}z^{9}-3a^{7}z^{7}+10a^{7}z^{5}-16a^{7}z^{3}+13a^{7}z-4a^{7}z^{-1}+2a^{6}z^{8}-7a^{6}z^{6}+17a^{6}z^{4}-15a^{6}z^{2}+5a^{6}+2a^{5}z^{7}-4a^{5}z^{5}+5a^{5}z^{3}+a^{5}z-a^{5}z^{-1}+a^{4}z^{6}+a^{4}z^{2}-a^{4}+3a^{3}z^{3}-5a^{3}z+a^{3}z^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums

\chi

(fixed

j

, alternation over

r

).

\		r
	\
j		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-2

2

-4

3

2

-1

-6

3

-8

3

0

-10

4

3

1

-12

2

3

1

-14

3

4

-1

-16

2

-18

1

3

-2

-20

0

-22

1

-1

Integral Khovanov Homology

(db, data source)

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