L11n128

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Planar diagram presentation	X₈₁₉₂ X_10,4,11,3 X_22,10,7,9 X₂₇₃₈ X_15,5,16,4 X_5,13,6,12 X_16,12,17,11 X_6,18,1,17 X_19,14,20,15 X_13,20,14,21 X_21,19,22,18
Gauss code	{1, -4, 2, 5, -6, -8}, {4, -1, 3, -2, 7, 6, -10, 9, -5, -7, 8, 11, -9, 10, -11, -3}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{2}v^{4}-3u^{2}v^{3}+2u^{2}v^{2}-uv^{4}+2uv^{3}-uv^{2}+2uv-u+2v^{2}-3v+1}{uv^{2}}}$ (db)
Jones polynomial	$4q^{9/2}-6q^{7/2}+6q^{5/2}-{\frac {1}{q^{5/2}}}-6q^{3/2}+{\frac {2}{q^{3/2}}}+q^{13/2}-3q^{11/2}+5{\sqrt {q}}-{\frac {4}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-3}-2z^{5}a^{-1}+5z^{5}a^{-3}-z^{5}a^{-5}+az^{3}-8z^{3}a^{-1}+8z^{3}a^{-3}-4z^{3}a^{-5}+3az-7za^{-1}+7za^{-3}-3za^{-5}+za^{-7}+az^{-1}-2a^{-1}z^{-1}+2a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$-z^{9}a^{-1}-z^{9}a^{-3}-5z^{8}a^{-2}-3z^{8}a^{-4}-2z^{8}-az^{7}-2z^{7}a^{-3}-3z^{7}a^{-5}+20z^{6}a^{-2}+10z^{6}a^{-4}-z^{6}a^{-6}+9z^{6}+5az^{5}+15z^{5}a^{-1}+20z^{5}a^{-3}+10z^{5}a^{-5}-20z^{4}a^{-2}-9z^{4}a^{-4}-11z^{4}-8az^{3}-25z^{3}a^{-1}-27z^{3}a^{-3}-13z^{3}a^{-5}-3z^{3}a^{-7}+6z^{2}a^{-2}+3z^{2}a^{-4}-z^{2}a^{-8}+4z^{2}+5az+13za^{-1}+13za^{-3}+7za^{-5}+2za^{-7}-a^{-2}-az^{-1}-2a^{-1}z^{-1}-2a^{-3}z^{-1}-a^{-5}z^{-1}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

χ

14

1

-1

12

2

10

2

1

-1

8

4

2

6

3

0

4

3

0

2

3

4

1

0

1

2

-1

-2

1

3

2

-4

1

-1

-6

1

Integral Khovanov Homology

(db, data source)

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