L11a50

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_18,7,19,8 X_4,19,1,20 X_12,6,13,5 X_10,4,11,3 X_22,14,5,13 X_14,22,15,21 X_20,12,21,11 X_16,9,17,10 X_2,16,3,15 X_8,17,9,18
Gauss code	{1, -10, 5, -3}, {4, -1, 2, -11, 9, -5, 8, -4, 6, -7, 10, -9, 11, -2, 3, -8, 7, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2(u-1)(v-1)\left(2v^{2}-3v+2\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-7q^{9/2}+{\frac {1}{q^{9/2}}}+11q^{7/2}-{\frac {4}{q^{7/2}}}-15q^{5/2}+{\frac {7}{q^{5/2}}}+18q^{3/2}-{\frac {12}{q^{3/2}}}-q^{13/2}+3q^{11/2}-18{\sqrt {q}}+{\frac {15}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-5}-za^{-5}-a^{-5}z^{-1}+z^{5}a^{-3}-a^{3}z^{3}+z^{3}a^{-3}+2za^{-3}+a^{3}z^{-1}+2a^{-3}z^{-1}+az^{5}+2z^{5}a^{-1}+3z^{3}a^{-1}-2az+za^{-1}-az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-2z^{3}a^{-7}+3z^{6}a^{-6}-5z^{4}a^{-6}+6z^{7}a^{-5}-14z^{5}a^{-5}+12z^{3}a^{-5}-5za^{-5}+a^{-5}z^{-1}+7z^{8}a^{-4}+a^{4}z^{6}-17z^{6}a^{-4}-2a^{4}z^{4}+19z^{4}a^{-4}-8z^{2}a^{-4}+a^{-4}+5z^{9}a^{-3}+4a^{3}z^{7}-7z^{7}a^{-3}-11a^{3}z^{5}-z^{5}a^{-3}+6a^{3}z^{3}+15z^{3}a^{-3}+a^{3}z-11za^{-3}-a^{3}z^{-1}+2a^{-3}z^{-1}+2z^{10}a^{-2}+6a^{2}z^{8}+5z^{8}a^{-2}-16a^{2}z^{6}-22z^{6}a^{-2}+10a^{2}z^{4}+31z^{4}a^{-2}-2a^{2}z^{2}-15z^{2}a^{-2}+a^{2}+3a^{-2}+5az^{9}+10z^{9}a^{-1}-10az^{7}-27z^{7}a^{-1}+4az^{5}+29z^{5}a^{-1}-3az^{3}-8z^{3}a^{-1}+3az-4za^{-1}-az^{-1}+a^{-1}z^{-1}+2z^{10}+4z^{8}-19z^{6}+19z^{4}-9z^{2}+2$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

2

-2

10

5

1

4

8

6

2

-4

6

9

5

4

9

6

-3

2

9

0

8

11

3

-2

4

7

-3

-4

3

8

5

-6

1

4

-3

-8

3

-10

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=0$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{9}$
$r=1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=2$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=4$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$