L11a19

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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_16,10,17,9 X_20,14,21,13 X_22,17,5,18 X_18,21,19,22 X_14,20,15,19 X_8,16,9,15 X₂₅₃₆ X_4,12,1,11
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 4, -2, 11, -3, 5, -8, 9, -4, 6, -7, 8, -5, 7, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{4}-4t(2)^{3}+4t(2)^{2}-4t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$16q^{9/2}-18q^{7/2}+17q^{5/2}-{\frac {1}{q^{5/2}}}-16q^{3/2}+{\frac {3}{q^{3/2}}}+q^{17/2}-3q^{15/2}+7q^{13/2}-12q^{11/2}+11{\sqrt {q}}-{\frac {7}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-7}+2za^{-7}+a^{-7}z^{-1}-2z^{5}a^{-5}-6z^{3}a^{-5}-7za^{-5}-3a^{-5}z^{-1}+z^{7}a^{-3}+4z^{5}a^{-3}+8z^{3}a^{-3}+9za^{-3}+3a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-6z^{3}a^{-1}+2az-6za^{-1}+az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-2}-z^{10}a^{-4}-3z^{9}a^{-1}-8z^{9}a^{-3}-5z^{9}a^{-5}-10z^{8}a^{-2}-16z^{8}a^{-4}-9z^{8}a^{-6}-3z^{8}-az^{7}+4z^{7}a^{-1}+10z^{7}a^{-3}-4z^{7}a^{-5}-9z^{7}a^{-7}+41z^{6}a^{-2}+49z^{6}a^{-4}+13z^{6}a^{-6}-6z^{6}a^{-8}+11z^{6}+4az^{5}+13z^{5}a^{-1}+25z^{5}a^{-3}+33z^{5}a^{-5}+14z^{5}a^{-7}-3z^{5}a^{-9}-42z^{4}a^{-2}-40z^{4}a^{-4}-4z^{4}a^{-6}+6z^{4}a^{-8}-z^{4}a^{-10}-13z^{4}-6az^{3}-25z^{3}a^{-1}-40z^{3}a^{-3}-36z^{3}a^{-5}-13z^{3}a^{-7}+2z^{3}a^{-9}+15z^{2}a^{-2}+11z^{2}a^{-4}-4z^{2}a^{-6}-4z^{2}a^{-8}+z^{2}a^{-10}+5z^{2}+4az+13za^{-1}+20za^{-3}+16za^{-5}+5za^{-7}-2a^{-2}+2a^{-6}+a^{-8}-az^{-1}-2a^{-1}z^{-1}-3a^{-3}z^{-1}-3a^{-5}z^{-1}-a^{-7}z^{-1}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

18

1

-1

16

2

14

5

1

-4

12

7

2

5

10

9

5

-4

8

9

7

2

6

8

9

1

4

8

9

-1

2

5

10

5

0

2

6

-4

-2

1

5

4

-4

2

-2

-6

1

Integral Khovanov Homology

(db, data source)

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