L11a112

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

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2(t(1)-1)(t(2)-1)\left(t(2)^{2}+1\right)\left(t(2)^{2}-t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{23/2}-3q^{21/2}+7q^{19/2}-11q^{17/2}+14q^{15/2}-15q^{13/2}+15q^{11/2}-13q^{9/2}+8q^{7/2}-6q^{5/2}+2q^{3/2}-{\sqrt {q}}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$z^{5}a^{-9}+3z^{3}a^{-9}+3za^{-9}+a^{-9}z^{-1}-z^{7}a^{-7}-4z^{5}a^{-7}-6z^{3}a^{-7}-5za^{-7}-a^{-7}z^{-1}-z^{7}a^{-5}-4z^{5}a^{-5}-5z^{3}a^{-5}-3za^{-5}-2a^{-5}z^{-1}+z^{5}a^{-3}+4z^{3}a^{-3}+5za^{-3}+2a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-14}-z^{2}a^{-14}+3z^{5}a^{-13}-2z^{3}a^{-13}+6z^{6}a^{-12}-7z^{4}a^{-12}+5z^{2}a^{-12}-a^{-12}+8z^{7}a^{-11}-12z^{5}a^{-11}+9z^{3}a^{-11}-za^{-11}+7z^{8}a^{-10}-9z^{6}a^{-10}+3z^{4}a^{-10}+z^{2}a^{-10}+4z^{9}a^{-9}-z^{7}a^{-9}-6z^{5}a^{-9}+2za^{-9}-a^{-9}z^{-1}+z^{10}a^{-8}+8z^{8}a^{-8}-23z^{6}a^{-8}+19z^{4}a^{-8}-13z^{2}a^{-8}+3a^{-8}+6z^{9}a^{-7}-13z^{7}a^{-7}+6z^{5}a^{-7}-5z^{3}a^{-7}+4za^{-7}-a^{-7}z^{-1}+z^{10}a^{-6}+3z^{8}a^{-6}-15z^{6}a^{-6}+13z^{4}a^{-6}-4z^{2}a^{-6}+2z^{9}a^{-5}-3z^{7}a^{-5}-8z^{5}a^{-5}+15z^{3}a^{-5}-6za^{-5}+2a^{-5}z^{-1}+2z^{8}a^{-4}-7z^{6}a^{-4}+5z^{4}a^{-4}+4z^{2}a^{-4}-3a^{-4}+z^{7}a^{-3}-5z^{5}a^{-3}+9z^{3}a^{-3}-7za^{-3}+2a^{-3}z^{-1}$ (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		\

-2

-1

0

1

2

3

4

5

6

7

8

9

χ

24

1

-1

22

2

20

5

1

-4

18

6

2

4

16

8

5

-3

14

7

6

1

12

8

0

10

5

7

-2

8

3

8

5

6

3

5

-2

4

1

5

4

2

1

-1

0

1

Integral Khovanov Homology

(db, data source)

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