L11a18

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)^{5}}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$q^{11/2}-4q^{9/2}+9q^{7/2}-15q^{5/2}+18q^{3/2}-21{\sqrt {q}}+{\frac {20}{\sqrt {q}}}-{\frac {17}{q^{3/2}}}+{\frac {12}{q^{5/2}}}-{\frac {7}{q^{7/2}}}+{\frac {3}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{5}z+a^{5}z^{-1}+z^{5}a^{-3}-3a^{3}z^{3}+2z^{3}a^{-3}-6a^{3}z+2za^{-3}-3a^{3}z^{-1}-z^{7}a^{-1}+3az^{5}-4z^{5}a^{-1}+9az^{3}-8z^{3}a^{-1}+10az-7za^{-1}+4az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{2}z^{10}-z^{10}-3a^{3}z^{9}-9az^{9}-6z^{9}a^{-1}-3a^{4}z^{8}-12a^{2}z^{8}-13z^{8}a^{-2}-22z^{8}-a^{5}z^{7}+3a^{3}z^{7}+7az^{7}-11z^{7}a^{-1}-14z^{7}a^{-3}+11a^{4}z^{6}+47a^{2}z^{6}+18z^{6}a^{-2}-9z^{6}a^{-4}+63z^{6}+4a^{5}z^{5}+16a^{3}z^{5}+41az^{5}+54z^{5}a^{-1}+21z^{5}a^{-3}-4z^{5}a^{-5}-14a^{4}z^{4}-51a^{2}z^{4}-5z^{4}a^{-2}+7z^{4}a^{-4}-z^{4}a^{-6}-50z^{4}-6a^{5}z^{3}-31a^{3}z^{3}-61az^{3}-51z^{3}a^{-1}-14z^{3}a^{-3}+z^{3}a^{-5}+7a^{4}z^{2}+20a^{2}z^{2}+z^{2}a^{-2}-2z^{2}a^{-4}+16z^{2}+4a^{5}z+18a^{3}z+28az+19za^{-1}+5za^{-3}-a^{4}-3a^{2}-a^{-2}-2-a^{5}z^{-1}-3a^{3}z^{-1}-4az^{-1}-2a^{-1}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

12

1

-1

10

3

8

6

1

-5

6

9

3

6

4

9

6

-3

2

12

9

3

0

10

11

1

-2

7

10

-3

-4

5

10

5

-6

2

7

-5

-8

1

5

4

-10

2

-2

-12

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-3$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-2$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=0$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{12}$
$r=1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=2$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$