L11a466

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(3)-1)\left(2t(2)t(3)^{2}-t(3)^{2}-4t(2)t(3)+4t(3)+t(2)-2\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{-9}-3q^{-8}+7q^{-7}-12q^{-6}+16q^{-5}-17q^{-4}+19q^{-3}-q^{2}-15q^{-2}+3q+12q^{-1}-6$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$z^{2}a^{8}+a^{8}-2z^{4}a^{6}-3z^{2}a^{6}+a^{6}z^{-2}-a^{6}+z^{6}a^{4}+z^{4}a^{4}-2z^{2}a^{4}-2a^{4}z^{-2}-4a^{4}+z^{6}a^{2}+3z^{4}a^{2}+6z^{2}a^{2}+a^{2}z^{-2}+5a^{2}-z^{4}-2z^{2}-1$ (db)
Kauffman polynomial	$z^{6}a^{10}-3z^{4}a^{10}+3z^{2}a^{10}-a^{10}+3z^{7}a^{9}-8z^{5}a^{9}+7z^{3}a^{9}-2za^{9}+4z^{8}a^{8}-6z^{6}a^{8}-3z^{4}a^{8}+6z^{2}a^{8}-a^{8}+3z^{9}a^{7}+2z^{7}a^{7}-17z^{5}a^{7}+15z^{3}a^{7}-6za^{7}+z^{10}a^{6}+8z^{8}a^{6}-15z^{6}a^{6}+6z^{2}a^{6}+a^{6}z^{-2}-3a^{6}+6z^{9}a^{5}-z^{7}a^{5}-15z^{5}a^{5}+12z^{3}a^{5}-za^{5}-2a^{5}z^{-1}+z^{10}a^{4}+8z^{8}a^{4}-9z^{6}a^{4}-9z^{4}a^{4}+16z^{2}a^{4}+2a^{4}z^{-2}-8a^{4}+3z^{9}a^{3}+4z^{7}a^{3}-10z^{5}a^{3}+3z^{3}a^{3}+5za^{3}-2a^{3}z^{-1}+4z^{8}a^{2}+2z^{6}a^{2}-15z^{4}a^{2}+18z^{2}a^{2}+a^{2}z^{-2}-8a^{2}+4z^{7}a-3z^{5}a-3z^{3}a+3za+3z^{6}-6z^{4}+5z^{2}-2+z^{5}a^{-1}-2z^{3}a^{-1}+za^{-1}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

5

1

-1

3

2

1

4

1

-3

-1

8

2

6

-3

9

6

-3

-5

10

6

4

-7

7

9

2

-9

9

10

-1

-11

5

9

4

-13

2

7

-5

-15

1

5

4

-17

2

-2

-19

1

Integral Khovanov Homology

(db, data source)

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