L11a350

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Planar diagram presentation	X_12,1,13,2 X₈₄₉₃ X_18,14,19,13 X_22,20,11,19 X_20,7,21,8 X_6,21,7,22 X_4,15,5,16 X_14,5,15,6 X_16,10,17,9 X_2,11,3,12 X_10,18,1,17
Gauss code	{1, -10, 2, -7, 8, -6, 5, -2, 9, -11}, {10, -1, 3, -8, 7, -9, 11, -3, 4, -5, 6, -4}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(2)^{3}t(1)^{3}-3t(2)^{2}t(1)^{3}+4t(2)t(1)^{3}-2t(1)^{3}-3t(2)^{3}t(1)^{2}+8t(2)^{2}t(1)^{2}-8t(2)t(1)^{2}+4t(1)^{2}+4t(2)^{3}t(1)-8t(2)^{2}t(1)+8t(2)t(1)-3t(1)-2t(2)^{3}+4t(2)^{2}-3t(2)+1}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{11/2}-4q^{9/2}+9q^{7/2}-15q^{5/2}+19q^{3/2}-21{\sqrt {q}}+{\frac {21}{\sqrt {q}}}-{\frac {18}{q^{3/2}}}+{\frac {12}{q^{5/2}}}-{\frac {8}{q^{7/2}}}+{\frac {3}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+3az^{5}-4z^{5}a^{-1}+z^{5}a^{-3}-3a^{3}z^{3}+8az^{3}-8z^{3}a^{-1}+2z^{3}a^{-3}+a^{5}z-5a^{3}z+6az-6za^{-1}+2za^{-3}+a^{5}z^{-1}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$-2a^{2}z^{10}-2z^{10}-4a^{3}z^{9}-12az^{9}-8z^{9}a^{-1}-3a^{4}z^{8}-6a^{2}z^{8}-14z^{8}a^{-2}-17z^{8}-a^{5}z^{7}+9a^{3}z^{7}+25az^{7}+z^{7}a^{-1}-14z^{7}a^{-3}+10a^{4}z^{6}+32a^{2}z^{6}+22z^{6}a^{-2}-9z^{6}a^{-4}+53z^{6}+4a^{5}z^{5}+a^{3}z^{5}-az^{5}+26z^{5}a^{-1}+20z^{5}a^{-3}-4z^{5}a^{-5}-10a^{4}z^{4}-32a^{2}z^{4}-10z^{4}a^{-2}+7z^{4}a^{-4}-z^{4}a^{-6}-40z^{4}-6a^{5}z^{3}-11a^{3}z^{3}-14az^{3}-21z^{3}a^{-1}-11z^{3}a^{-3}+z^{3}a^{-5}+2a^{4}z^{2}+9a^{2}z^{2}+2z^{2}a^{-2}-2z^{2}a^{-4}+11z^{2}+4a^{5}z+5a^{3}z+4az+5za^{-1}+2za^{-3}+a^{4}-a^{5}z^{-1}-a^{3}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

12

1

-1

10

3

8

6

1

-5

6

9

3

6

4

10

6

-4

2

11

9

2

0

11

0

-2

7

10

-3

-4

5

11

6

-6

3

7

-4

-8

1

6

5

-10

2

-2

-12

1

Integral Khovanov Homology

(db, data source)

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