L11a406

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Link Presentations

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2uv^{2}w-2uv^{2}+2uvw^{2}-7uvw+4uv-2uw^{2}+4uw-2u+2v^{2}w^{2}-4v^{2}w+2v^{2}-4vw^{2}+7vw-2v+2w^{2}-2w}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$-q^{-10}+3q^{-9}-6q^{-8}+10q^{-7}-13q^{-6}+16q^{-5}-15q^{-4}+15q^{-3}-10q^{-2}+q+7q^{-1}-3$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{10}+3a^{8}z^{2}+2a^{8}-2a^{6}z^{4}-a^{6}z^{2}+a^{6}z^{-2}-3a^{4}z^{4}-4a^{4}z^{2}-2a^{4}z^{-2}-4a^{4}-a^{2}z^{4}+2a^{2}z^{2}+a^{2}z^{-2}+3a^{2}+z^{2}$ (db)
Kauffman polynomial	$z^{7}a^{11}-4z^{5}a^{11}+5z^{3}a^{11}-2za^{11}+3z^{8}a^{10}-12z^{6}a^{10}+15z^{4}a^{10}-7z^{2}a^{10}+2a^{10}+3z^{9}a^{9}-7z^{7}a^{9}-3z^{5}a^{9}+12z^{3}a^{9}-4za^{9}+z^{10}a^{8}+7z^{8}a^{8}-34z^{6}a^{8}+42z^{4}a^{8}-21z^{2}a^{8}+4a^{8}+7z^{9}a^{7}-15z^{7}a^{7}+3z^{5}a^{7}+2z^{3}a^{7}+z^{10}a^{6}+10z^{8}a^{6}-30z^{6}a^{6}+23z^{4}a^{6}-7z^{2}a^{6}+a^{6}z^{-2}-2a^{6}+4z^{9}a^{5}-8z^{5}a^{5}-2z^{3}a^{5}+6za^{5}-2a^{5}z^{-1}+6z^{8}a^{4}-2z^{6}a^{4}-13z^{4}a^{4}+16z^{2}a^{4}+2a^{4}z^{-2}-7a^{4}+7z^{7}a^{3}-7z^{5}a^{3}+z^{3}a^{3}+4za^{3}-2a^{3}z^{-1}+6z^{6}a^{2}-8z^{4}a^{2}+8z^{2}a^{2}+a^{2}z^{-2}-4a^{2}+3z^{5}a-2z^{3}a+z^{4}-z^{2}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

3

1

2

-2

-1

5

1

4

-3

6

3

-3

-5

9

4

5

-7

8

0

-9

8

7

1

-11

6

9

3

-13

4

7

-3

-15

2

6

4

-17

1

4

-3

-19

2

-21

1

-1

Integral Khovanov Homology

(db, data source)

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