L11a87

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

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{5}-5uv^{4}+10uv^{3}-12uv^{2}+7uv-2u-2v^{5}+7v^{4}-12v^{3}+10v^{2}-5v+1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {14}{q^{9/2}}}-q^{7/2}+{\frac {19}{q^{7/2}}}+5q^{5/2}-{\frac {24}{q^{5/2}}}-11q^{3/2}+{\frac {24}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {3}{q^{13/2}}}+{\frac {7}{q^{11/2}}}+17{\sqrt {q}}-{\frac {22}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}(-z)-a^{7}z^{-1}+3a^{5}z^{3}+6a^{5}z+4a^{5}z^{-1}-3a^{3}z^{5}-8a^{3}z^{3}-9a^{3}z-4a^{3}z^{-1}+az^{7}+3az^{5}-z^{5}a^{-1}+4az^{3}-z^{3}a^{-1}+2az+az^{-1}$ (db)
Kauffman polynomial	$-2a^{4}z^{10}-2a^{2}z^{10}-4a^{5}z^{9}-12a^{3}z^{9}-8az^{9}-4a^{6}z^{8}-9a^{4}z^{8}-18a^{2}z^{8}-13z^{8}-3a^{7}z^{7}-a^{5}z^{7}+12a^{3}z^{7}-az^{7}-11z^{7}a^{-1}-a^{8}z^{6}+4a^{6}z^{6}+19a^{4}z^{6}+37a^{2}z^{6}-5z^{6}a^{-2}+18z^{6}+8a^{7}z^{5}+18a^{5}z^{5}+14a^{3}z^{5}+20az^{5}+15z^{5}a^{-1}-z^{5}a^{-3}+3a^{8}z^{4}+7a^{6}z^{4}-a^{4}z^{4}-13a^{2}z^{4}+4z^{4}a^{-2}-4z^{4}-8a^{7}z^{3}-25a^{5}z^{3}-25a^{3}z^{3}-12az^{3}-4z^{3}a^{-1}-3a^{8}z^{2}-11a^{6}z^{2}-15a^{4}z^{2}-8a^{2}z^{2}-z^{2}+4a^{7}z+16a^{5}z+15a^{3}z+3az+a^{8}+4a^{6}+7a^{4}+4a^{2}+1-a^{7}z^{-1}-4a^{5}z^{-1}-4a^{3}z^{-1}-az^{-1}$ (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

4

-4

4

7

1

6

2

10

4

-6

0

12

7

5

-2

13

11

-2

-4

11

0

-6

8

13

5

-8

6

11

-5

-10

2

9

7

-12

1

5

-4

-14

2

-16

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-4$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{6}$
$r=-3$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-2$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=-1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=0$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{12}$
$r=1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$