L11n434

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Planar diagram presentation	X₈₁₉₂ X_14,4,15,3 X_12,14,7,13 X₂₇₃₈ X_22,10,13,9 X_6,22,1,21 X_15,20,16,21 X_5,17,6,16 X_18,11,19,12 X_10,17,11,18 X_19,5,20,4
Gauss code	{1, -4, 2, 11, -8, -6}, {4, -1, 5, -10, 9, -3}, {3, -2, -7, 8, 10, -9, -11, 7, 6, -5}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2u^{2}vw-u^{2}v-u^{2}w+u^{2}+uv^{2}w^{3}-uv^{2}w^{2}+uv^{2}w-uv^{2}-2uvw^{3}+3uvw^{2}-3uvw+2uv+uw^{3}-uw^{2}+uw-u-v^{2}w^{3}+v^{2}w^{2}+vw^{3}-2vw^{2}}{uvw^{3/2}}}$ (db)
Jones polynomial	$q^{6}-3q^{5}+6q^{4}-7q^{3}-q^{-3}+10q^{2}+4q^{-2}-9q-6q^{-1}+9$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-2}-4z^{4}a^{-2}+z^{4}a^{-4}+2z^{4}-a^{2}z^{2}-7z^{2}a^{-2}+2z^{2}a^{-4}+4z^{2}-5a^{-2}+2a^{-4}+3-2a^{-2}z^{-2}+a^{-4}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$z^{6}a^{-6}-3z^{4}a^{-6}+2z^{2}a^{-6}+3z^{7}a^{-5}-9z^{5}a^{-5}+5z^{3}a^{-5}+4z^{8}a^{-4}-13z^{6}a^{-4}+13z^{4}a^{-4}-11z^{2}a^{-4}-a^{-4}z^{-2}+6a^{-4}+2z^{9}a^{-3}-2z^{7}a^{-3}-7z^{5}a^{-3}+a^{3}z^{3}+8z^{3}a^{-3}-a^{3}z-6za^{-3}+2a^{-3}z^{-1}+8z^{8}a^{-2}-30z^{6}a^{-2}+4a^{2}z^{4}+44z^{4}a^{-2}-4a^{2}z^{2}-33z^{2}a^{-2}-2a^{-2}z^{-2}+a^{2}+12a^{-2}+2z^{9}a^{-1}+2az^{7}-3z^{7}a^{-1}-4az^{5}-2z^{5}a^{-1}+8az^{3}+10z^{3}a^{-1}-3az-8za^{-1}+2a^{-1}z^{-1}+4z^{8}-16z^{6}+32z^{4}-24z^{2}-z^{-2}+8$ (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		\

-3

-2

-1

0

1

2

3

4

5

6

χ

13

1

11

2

-2

9

4

1

3

7

4

3

-1

5

6

3

4

5

1

5

0

-1

2

5

3

-3

2

4

-2

-5

3

-7

1

-1

Integral Khovanov Homology

(db, data source)

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