L11a223

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_14,9,15,10 X₄₇₅₈ X_16,6,17,5 X_18,16,19,15 X_6,18,1,17 X_22,19,7,20 X_20,12,21,11 X_10,22,11,21 X_2,14,3,13 X_12,4,13,3
Gauss code	{1, -10, 11, -3, 4, -6}, {3, -1, 2, -9, 8, -11, 10, -2, 5, -4, 6, -5, 7, -8, 9, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)^{2}t(2)^{4}-2t(1)t(2)^{4}+t(2)^{4}-4t(1)^{2}t(2)^{3}+10t(1)t(2)^{3}-5t(2)^{3}+7t(1)^{2}t(2)^{2}-17t(1)t(2)^{2}+7t(2)^{2}-5t(1)^{2}t(2)+10t(1)t(2)-4t(2)+t(1)^{2}-2t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-16q^{9/2}+21q^{7/2}-{\frac {1}{q^{7/2}}}-25q^{5/2}+{\frac {4}{q^{5/2}}}+25q^{3/2}-{\frac {10}{q^{3/2}}}+q^{15/2}-4q^{13/2}+9q^{11/2}-22{\sqrt {q}}+{\frac {16}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+az^{5}-4z^{5}a^{-1}+3z^{5}a^{-3}+2az^{3}-9z^{3}a^{-1}+7z^{3}a^{-3}-3z^{3}a^{-5}+3az-7za^{-1}+7za^{-3}-3za^{-5}+za^{-7}+az^{-1}-2a^{-1}z^{-1}+2a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-2}-2z^{10}a^{-4}-7z^{9}a^{-1}-13z^{9}a^{-3}-6z^{9}a^{-5}-20z^{8}a^{-2}-16z^{8}a^{-4}-7z^{8}a^{-6}-11z^{8}-9az^{7}-6z^{7}a^{-1}+9z^{7}a^{-3}+2z^{7}a^{-5}-4z^{7}a^{-7}-4a^{2}z^{6}+45z^{6}a^{-2}+41z^{6}a^{-4}+14z^{6}a^{-6}-z^{6}a^{-8}+15z^{6}-a^{3}z^{5}+14az^{5}+33z^{5}a^{-1}+27z^{5}a^{-3}+18z^{5}a^{-5}+9z^{5}a^{-7}+4a^{2}z^{4}-28z^{4}a^{-2}-26z^{4}a^{-4}-8z^{4}a^{-6}+2z^{4}a^{-8}-8z^{4}+a^{3}z^{3}-11az^{3}-32z^{3}a^{-1}-32z^{3}a^{-3}-19z^{3}a^{-5}-7z^{3}a^{-7}-a^{2}z^{2}+6z^{2}a^{-2}+5z^{2}a^{-4}+z^{2}a^{-6}-z^{2}a^{-8}+2z^{2}+5az+13za^{-1}+13za^{-3}+7za^{-5}+2za^{-7}-a^{-2}-az^{-1}-2a^{-1}z^{-1}-2a^{-3}z^{-1}-a^{-5}z^{-1}$ (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

16

1

-1

14

3

12

6

1

-5

10

3

7

8

11

6

-5

6

14

10

4

12

0

2

10

13

-3

0

7

13

6

-2

3

9

-6

-4

1

7

6

-6

3

-3

-8

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-4$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=0$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{10}$
$r=1$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=2$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{14}$
$r=3$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=4$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=6$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=7$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$