L11a234

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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_22,12,7,11 X_20,17,21,18 X_16,21,17,22 X_14,10,15,9 X_10,16,11,15 X_4,20,5,19 X₂₇₃₈ X_6,14,1,13
Gauss code	{1, -10, 2, -9, 3, -11}, {10, -1, 7, -8, 4, -2, 11, -7, 8, -6, 5, -3, 9, -5, 6, -4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)^{2}t(2)^{4}-3t(1)t(2)^{4}+2t(2)^{4}-3t(1)^{2}t(2)^{3}+8t(1)t(2)^{3}-4t(2)^{3}+4t(1)^{2}t(2)^{2}-9t(1)t(2)^{2}+4t(2)^{2}-4t(1)^{2}t(2)+8t(1)t(2)-3t(2)+2t(1)^{2}-3t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{17/2}-3q^{15/2}+7q^{13/2}-12q^{11/2}+16q^{9/2}-19q^{7/2}+18q^{5/2}-17q^{3/2}+12{\sqrt {q}}-{\frac {8}{\sqrt {q}}}+{\frac {4}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-7}+2za^{-7}+a^{-7}z^{-1}-2z^{5}a^{-5}-6z^{3}a^{-5}-7za^{-5}-3a^{-5}z^{-1}+z^{7}a^{-3}+4z^{5}a^{-3}+8z^{3}a^{-3}+8za^{-3}+2a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-5z^{3}a^{-1}+az-3za^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-10}-z^{2}a^{-10}+3z^{5}a^{-9}-2z^{3}a^{-9}+6z^{6}a^{-8}-6z^{4}a^{-8}+4z^{2}a^{-8}-a^{-8}+9z^{7}a^{-7}-14z^{5}a^{-7}+13z^{3}a^{-7}-5za^{-7}+a^{-7}z^{-1}+9z^{8}a^{-6}-12z^{6}a^{-6}+z^{4}a^{-6}+7z^{2}a^{-6}-3a^{-6}+6z^{9}a^{-5}-z^{7}a^{-5}-23z^{5}a^{-5}+27z^{3}a^{-5}-13za^{-5}+3a^{-5}z^{-1}+2z^{10}a^{-4}+11z^{8}a^{-4}-38z^{6}a^{-4}+26z^{4}a^{-4}-2z^{2}a^{-4}-3a^{-4}+11z^{9}a^{-3}-25z^{7}a^{-3}+3z^{5}a^{-3}+15z^{3}a^{-3}-10za^{-3}+2a^{-3}z^{-1}+2z^{10}a^{-2}+6z^{8}a^{-2}-34z^{6}a^{-2}+32z^{4}a^{-2}-7z^{2}a^{-2}+5z^{9}a^{-1}+az^{7}-14z^{7}a^{-1}-3az^{5}+6z^{5}a^{-1}+3az^{3}+6z^{3}a^{-1}-az-3za^{-1}+4z^{8}-14z^{6}+14z^{4}-3z^{2}$ (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		\

-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

10

7

3

6

9

10

1

4

8

9

-1

2

5

10

5

0

3

7

-4

-2

1

5

4

-4

3

-3

-6

1

Integral Khovanov Homology

(db, data source)

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