L11a539

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

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(w-1)(x-1)\left(uvx^{2}-uvx+uv-ux^{2}+2ux-2u-2vx^{2}+2vx-v+x^{2}-x+1\right)}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}x^{3/2}}}$ (db)
Jones polynomial	$-4q^{9/2}+{\frac {2}{q^{9/2}}}+10q^{7/2}-{\frac {7}{q^{7/2}}}-16q^{5/2}+{\frac {10}{q^{5/2}}}+18q^{3/2}-{\frac {18}{q^{3/2}}}+q^{11/2}-{\frac {1}{q^{11/2}}}-22{\sqrt {q}}+{\frac {19}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+3az^{5}-4z^{5}a^{-1}+z^{5}a^{-3}-3a^{3}z^{3}+10az^{3}-9z^{3}a^{-1}+2z^{3}a^{-3}+a^{5}z-8a^{3}z+15az-11za^{-1}+3za^{-3}+2a^{5}z^{-1}-8a^{3}z^{-1}+11az^{-1}-6a^{-1}z^{-1}+a^{-3}z^{-1}+a^{5}z^{-3}-3a^{3}z^{-3}+3az^{-3}-a^{-1}z^{-3}$ (db)
Kauffman polynomial	$z^{4}a^{-6}+a^{5}z^{7}-5a^{5}z^{5}+4z^{5}a^{-5}+10a^{5}z^{3}-a^{5}z^{-3}-10a^{5}z+5a^{5}z^{-1}+2a^{4}z^{8}-6a^{4}z^{6}+10z^{6}a^{-4}+3a^{4}z^{4}-8z^{4}a^{-4}+7a^{4}z^{2}+4z^{2}a^{-4}+3a^{4}z^{-2}-9a^{4}-a^{-4}+2a^{3}z^{9}+a^{3}z^{7}+16z^{7}a^{-3}-23a^{3}z^{5}-25z^{5}a^{-3}+43a^{3}z^{3}+17z^{3}a^{-3}-3a^{3}z^{-3}-34a^{3}z-9za^{-3}+14a^{3}z^{-1}+2a^{-3}z^{-1}+a^{2}z^{10}+8a^{2}z^{8}+14z^{8}a^{-2}-27a^{2}z^{6}-17z^{6}a^{-2}+14a^{2}z^{4}-6z^{4}a^{-2}+19a^{2}z^{2}+15z^{2}a^{-2}+6a^{2}z^{-2}-21a^{2}-6a^{-2}+8az^{9}+6z^{9}a^{-1}-3az^{7}+13z^{7}a^{-1}-48az^{5}-59z^{5}a^{-1}+77az^{3}+61z^{3}a^{-1}-3az^{-3}-a^{-1}z^{-3}-50az-35za^{-1}+18az^{-1}+11a^{-1}z^{-1}+z^{10}+20z^{8}-48z^{6}+14z^{4}+23z^{2}+3z^{-2}-18$ (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

4

5

χ

12

1

-1

10

3

8

7

1

-6

6

9

3

6

4

9

7

-2

2

13

9

4

0

12

15

3

-2

6

7

-1

-4

4

12

8

-6

3

6

-3

-8

1

6

5

-10

1

-1

-12

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{3}$
$r=-3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-2$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=0$	${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{13}$
$r=1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=2$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$