L11a100

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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_20,11,21,12 X_8,17,9,18 X_10,19,11,20 X_4,21,1,22 X_14,6,15,5 X_12,4,13,3 X_22,14,5,13 X_2,16,3,15
Gauss code	{1, -11, 9, -7}, {8, -1, 2, -5, 3, -6, 4, -9, 10, -8, 11, -2, 5, -3, 6, -4, 7, -10}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)\left(v^{2}-v+1\right)^{2}\left(v^{2}+v+1\right)}{{\sqrt {u}}v^{7/2}}}$ (db)
Jones polynomial	$-q^{5/2}+3q^{3/2}-5{\sqrt {q}}+{\frac {6}{\sqrt {q}}}-{\frac {10}{q^{3/2}}}+{\frac {10}{q^{5/2}}}-{\frac {11}{q^{7/2}}}+{\frac {10}{q^{9/2}}}-{\frac {7}{q^{11/2}}}+{\frac {5}{q^{13/2}}}-{\frac {3}{q^{15/2}}}+{\frac {1}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{3}z^{9}-a^{5}z^{7}+7a^{3}z^{7}-az^{7}-5a^{5}z^{5}+17a^{3}z^{5}-5az^{5}-7a^{5}z^{3}+17a^{3}z^{3}-7az^{3}-3a^{5}z+7a^{3}z-4az-a^{5}z^{-1}+3a^{3}z^{-1}-2az^{-1}$ (db)
Kauffman polynomial	$a^{10}z^{4}-a^{10}z^{2}+3a^{9}z^{5}-4a^{9}z^{3}+4a^{8}z^{6}-5a^{8}z^{4}+a^{8}z^{2}+4a^{7}z^{7}-5a^{7}z^{5}+2a^{7}z^{3}+4a^{6}z^{8}-8a^{6}z^{6}+7a^{6}z^{4}-2a^{6}z^{2}+a^{6}+4a^{5}z^{9}-14a^{5}z^{7}+23a^{5}z^{5}-18a^{5}z^{3}+6a^{5}z-a^{5}z^{-1}+2a^{4}z^{10}-4a^{4}z^{8}-3a^{4}z^{6}+11a^{4}z^{4}-8a^{4}z^{2}+3a^{4}+8a^{3}z^{9}-38a^{3}z^{7}+63a^{3}z^{5}-46a^{3}z^{3}+13a^{3}z-3a^{3}z^{-1}+2a^{2}z^{10}-5a^{2}z^{8}-4a^{2}z^{6}+11a^{2}z^{4}-6a^{2}z^{2}+3a^{2}+4az^{9}-19az^{7}+z^{7}a^{-1}+28az^{5}-4z^{5}a^{-1}-19az^{3}+3z^{3}a^{-1}+7az-2az^{-1}+3z^{8}-13z^{6}+13z^{4}-2z^{2}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

6

1

4

2

-2

2

3

1

2

0

3

2

-1

-2

7

3

4

-4

5

0

-6

6

5

1

-8

4

5

1

-10

3

6

-3

-12

2

4

2

-14

1

3

-2

-16

2

-18

1

-1

Integral Khovanov Homology

(db, data source)

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