L11a69

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_14,10,15,9 X_22,20,5,19 X_20,11,21,12 X_10,21,11,22 X_18,15,19,16 X_16,7,17,8 X_8,17,9,18 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 8, -9, 3, -6, 5, -2, 11, -3, 7, -8, 9, -7, 4, -5, 6, -4}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)\left(v^{4}-4v^{3}+7v^{2}-4v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-q^{7/2}+4q^{5/2}-9q^{3/2}+14{\sqrt {q}}-{\frac {20}{\sqrt {q}}}+{\frac {22}{q^{3/2}}}-{\frac {22}{q^{5/2}}}+{\frac {18}{q^{7/2}}}-{\frac {14}{q^{9/2}}}+{\frac {8}{q^{11/2}}}-{\frac {3}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}(-z)-a^{7}z^{-1}+3a^{5}z^{3}+5a^{5}z+3a^{5}z^{-1}-3a^{3}z^{5}-8a^{3}z^{3}-8a^{3}z-3a^{3}z^{-1}+az^{7}+4az^{5}-z^{5}a^{-1}+8az^{3}-2z^{3}a^{-1}+6az+2az^{-1}-2za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{4}z^{10}-a^{2}z^{10}-4a^{5}z^{9}-9a^{3}z^{9}-5az^{9}-5a^{6}z^{8}-16a^{4}z^{8}-20a^{2}z^{8}-9z^{8}-3a^{7}z^{7}-2a^{5}z^{7}-9az^{7}-8z^{7}a^{-1}-a^{8}z^{6}+9a^{6}z^{6}+41a^{4}z^{6}+47a^{2}z^{6}-4z^{6}a^{-2}+12z^{6}+7a^{7}z^{5}+21a^{5}z^{5}+36a^{3}z^{5}+36az^{5}+13z^{5}a^{-1}-z^{5}a^{-3}+3a^{8}z^{4}-3a^{6}z^{4}-32a^{4}z^{4}-37a^{2}z^{4}+5z^{4}a^{-2}-6z^{4}-6a^{7}z^{3}-22a^{5}z^{3}-41a^{3}z^{3}-35az^{3}-9z^{3}a^{-1}+z^{3}a^{-3}-3a^{8}z^{2}-2a^{6}z^{2}+9a^{4}z^{2}+11a^{2}z^{2}-z^{2}a^{-2}+2z^{2}+3a^{7}z+12a^{5}z+18a^{3}z+13az+4za^{-1}+a^{8}+2a^{6}-2a^{2}-a^{7}z^{-1}-3a^{5}z^{-1}-3a^{3}z^{-1}-2az^{-1}-a^{-1}z^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

3

-3

4

6

1

5

2

8

3

-5

0

12

6

-2

12

10

-2

-4

10

0

-6

8

12

4

-8

6

10

-4

-10

2

8

6

-12

1

6

-5

-14

2

-16

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-4$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-3$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-2$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=0$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{12}$
$r=1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$