L11n275

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Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₃₈₄₉ X_2,14,3,13 X_14,7,15,8 X_9,18,10,19 X_17,11,18,22 X_11,21,12,20 X_21,17,22,16 X_15,1,16,4 X_19,10,20,5
Gauss code	{1, -4, -3, 10}, {-2, -1, 5, 3, -6, 11}, {-8, 2, 4, -5, -10, 9, -7, 6, -11, 8, -9, 7}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {uvw^{4}-3uvw^{3}+3uvw^{2}-uvw+2uw^{3}-2uw^{2}+uw-v^{2}w^{3}+2v^{2}w^{2}-2v^{2}w+vw^{3}-3vw^{2}+3vw-v}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$-2q^{3}+4q^{2}-6q+9-8q^{-1}+9q^{-2}-6q^{-3}+5q^{-4}-2q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-a^{2}z^{6}+a^{4}z^{4}-5a^{2}z^{4}+3z^{4}+3a^{4}z^{2}-12a^{2}z^{2}-2z^{2}a^{-2}+10z^{2}+4a^{4}-13a^{2}-3a^{-2}+12+2a^{4}z^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}+4z^{-2}$ (db)
Kauffman polynomial	$a^{3}z^{9}+az^{9}+3a^{4}z^{8}+6a^{2}z^{8}+3z^{8}+2a^{5}z^{7}+3a^{3}z^{7}+4az^{7}+3z^{7}a^{-1}+a^{6}z^{6}-11a^{4}z^{6}-22a^{2}z^{6}+z^{6}a^{-2}-9z^{6}-6a^{5}z^{5}-20a^{3}z^{5}-23az^{5}-9z^{5}a^{-1}-4a^{6}z^{4}+15a^{4}z^{4}+36a^{2}z^{4}+2z^{4}a^{-2}+19z^{4}+3a^{5}z^{3}+28a^{3}z^{3}+43az^{3}+21z^{3}a^{-1}+3z^{3}a^{-3}+4a^{6}z^{2}-15a^{4}z^{2}-36a^{2}z^{2}-4z^{2}a^{-2}-21z^{2}-18a^{3}z-33az-19za^{-1}-4za^{-3}-a^{6}+8a^{4}+20a^{2}+3a^{-2}+15+5a^{3}z^{-1}+9az^{-1}+5a^{-1}z^{-1}+a^{-3}z^{-1}-2a^{4}z^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}-4z^{-2}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

7

2

-2

5

2

3

4

2

-2

1

5

2

3

-1

5

6

1

-3

4

3

1

-5

3

6

3

-7

2

3

-1

-9

3

-11

1

2

-1

-13

1

Integral Khovanov Homology

(db, data source)

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