L11a232

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_12,4,13,3 X_18,6,19,5 X_22,16,7,15 X_16,11,17,12 X_20,17,21,18 X_10,21,11,22 X_14,10,15,9 X_4,20,5,19 X₂₇₃₈ X_6,14,1,13
Gauss code	{1, -10, 2, -9, 3, -11}, {10, -1, 8, -7, 5, -2, 11, -8, 4, -5, 6, -3, 9, -6, 7, -4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)^{2}t(2)^{4}-3t(1)t(2)^{4}+2t(2)^{4}-4t(1)^{2}t(2)^{3}+11t(1)t(2)^{3}-6t(2)^{3}+7t(1)^{2}t(2)^{2}-17t(1)t(2)^{2}+7t(2)^{2}-6t(1)^{2}t(2)+11t(1)t(2)-4t(2)+2t(1)^{2}-3t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-10q^{9/2}+{\frac {1}{q^{9/2}}}+17q^{7/2}-{\frac {5}{q^{7/2}}}-24q^{5/2}+{\frac {11}{q^{5/2}}}+27q^{3/2}-{\frac {18}{q^{3/2}}}-q^{13/2}+4q^{11/2}-28{\sqrt {q}}+{\frac {24}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-5}-za^{-5}-a^{-5}z^{-1}+2z^{5}a^{-3}-a^{3}z^{3}+4z^{3}a^{-3}+5za^{-3}+2a^{-3}z^{-1}-z^{7}a^{-1}+2az^{5}-3z^{5}a^{-1}+3az^{3}-6z^{3}a^{-1}+2az-5za^{-1}+az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-z^{3}a^{-7}+4z^{6}a^{-6}-4z^{4}a^{-6}+z^{2}a^{-6}+9z^{7}a^{-5}-13z^{5}a^{-5}+10z^{3}a^{-5}-4za^{-5}+a^{-5}z^{-1}+12z^{8}a^{-4}+a^{4}z^{6}-17z^{6}a^{-4}-a^{4}z^{4}+11z^{4}a^{-4}-3z^{2}a^{-4}+9z^{9}a^{-3}+5a^{3}z^{7}-9a^{3}z^{5}-24z^{5}a^{-3}+4a^{3}z^{3}+27z^{3}a^{-3}-12za^{-3}+2a^{-3}z^{-1}+3z^{10}a^{-2}+10a^{2}z^{8}+21z^{8}a^{-2}-21a^{2}z^{6}-53z^{6}a^{-2}+12a^{2}z^{4}+38z^{4}a^{-2}-a^{2}z^{2}-10z^{2}a^{-2}+a^{-2}+9az^{9}+18z^{9}a^{-1}-9az^{7}-23z^{7}a^{-1}-12az^{5}-13z^{5}a^{-1}+13az^{3}+25z^{3}a^{-1}-4az-12za^{-1}+az^{-1}+2a^{-1}z^{-1}+3z^{10}+19z^{8}-54z^{6}+36z^{4}-7z^{2}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

3

-3

10

7

1

6

8

10

3

-7

6

14

7

4

14

11

-3

2

14

13

1

0

11

15

4

-2

7

13

-6

-4

4

11

7

-6

1

7

-6

-8

4

-10

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-1$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=0$	${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{14}$
$r=1$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{14}$	${\mathbb {Z} }^{14}$
$r=2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{14}$
$r=3$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$