L11a51

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)\left(v^{2}+1\right)\left(v^{4}-v^{3}+v^{2}-v+1\right)}{{\sqrt {u}}v^{7/2}}}$ (db)
Jones polynomial	$-q^{13/2}+3q^{11/2}-5q^{9/2}+8q^{7/2}-11q^{5/2}+12q^{3/2}-13{\sqrt {q}}+{\frac {10}{\sqrt {q}}}-{\frac {8}{q^{3/2}}}+{\frac {5}{q^{5/2}}}-{\frac {3}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{9}a^{-1}-az^{7}+7z^{7}a^{-1}-z^{7}a^{-3}-5az^{5}+17z^{5}a^{-1}-5z^{5}a^{-3}-7az^{3}+16z^{3}a^{-1}-7z^{3}a^{-3}-2az+4za^{-1}-2za^{-3}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-2}-2z^{10}-4az^{9}-8z^{9}a^{-1}-4z^{9}a^{-3}-4a^{2}z^{8}+4z^{8}a^{-2}-4z^{8}a^{-4}+4z^{8}-3a^{3}z^{7}+15az^{7}+36z^{7}a^{-1}+14z^{7}a^{-3}-4z^{7}a^{-5}-a^{4}z^{6}+13a^{2}z^{6}+9z^{6}a^{-4}-3z^{6}a^{-6}+2z^{6}+10a^{3}z^{5}-21az^{5}-65z^{5}a^{-1}-24z^{5}a^{-3}+9z^{5}a^{-5}-z^{5}a^{-7}+3a^{4}z^{4}-9a^{2}z^{4}-11z^{4}a^{-2}-6z^{4}a^{-4}+7z^{4}a^{-6}-10z^{4}-6a^{3}z^{3}+16az^{3}+44z^{3}a^{-1}+16z^{3}a^{-3}-4z^{3}a^{-5}+2z^{3}a^{-7}-a^{4}z^{2}+a^{2}z^{2}+5z^{2}a^{-2}-2z^{2}a^{-6}+5z^{2}-4az-8za^{-1}-4za^{-3}+1-az^{-1}-a^{-1}z^{-1}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

2

-2

10

3

1

2

8

5

2

-3

6

3

4

6

5

-1

2

7

6

1

0

5

8

3

-2

3

5

-2

-4

2

5

3

-6

1

3

-2

-8

2

-10

1

-1

Integral Khovanov Homology

(db, data source)

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