L11a214

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)^{2}t(2)^{4}-3t(1)t(2)^{4}-4t(1)^{2}t(2)^{3}+9t(1)t(2)^{3}-4t(2)^{3}+6t(1)^{2}t(2)^{2}-13t(1)t(2)^{2}+6t(2)^{2}-4t(1)^{2}t(2)+9t(1)t(2)-4t(2)-3t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{9/2}-4q^{7/2}+9q^{5/2}-14q^{3/2}+19{\sqrt {q}}-{\frac {22}{\sqrt {q}}}+{\frac {21}{q^{3/2}}}-{\frac {19}{q^{5/2}}}+{\frac {13}{q^{7/2}}}-{\frac {8}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-2a^{3}z^{5}+3az^{5}-2z^{5}a^{-1}+a^{5}z^{3}-5a^{3}z^{3}+3az^{3}-4z^{3}a^{-1}+z^{3}a^{-3}+2a^{5}z-4a^{3}z-az-za^{-1}+za^{-3}+a^{5}z^{-1}-2az^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2a^{2}z^{10}-2z^{10}-6a^{3}z^{9}-12az^{9}-6z^{9}a^{-1}-7a^{4}z^{8}-12a^{2}z^{8}-7z^{8}a^{-2}-12z^{8}-6a^{5}z^{7}+5a^{3}z^{7}+22az^{7}+7z^{7}a^{-1}-4z^{7}a^{-3}-3a^{6}z^{6}+8a^{4}z^{6}+34a^{2}z^{6}+16z^{6}a^{-2}-z^{6}a^{-4}+40z^{6}-a^{7}z^{5}+10a^{5}z^{5}-a^{3}z^{5}-14az^{5}+7z^{5}a^{-1}+9z^{5}a^{-3}+4a^{6}z^{4}-2a^{4}z^{4}-35a^{2}z^{4}-10z^{4}a^{-2}+2z^{4}a^{-4}-41z^{4}+2a^{7}z^{3}-9a^{5}z^{3}+9az^{3}-7z^{3}a^{-1}-5z^{3}a^{-3}-a^{6}z^{2}-2a^{4}z^{2}+16a^{2}z^{2}+4z^{2}a^{-2}-z^{2}a^{-4}+22z^{2}-a^{7}z+5a^{5}z-7az+za^{-3}+a^{4}-3a^{2}-2a^{-2}-5-a^{5}z^{-1}+2az^{-1}+a^{-1}z^{-1}$ (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

10

1

-1

8

3

6

1

-5

4

8

3

5

2

11

6

-5

0

11

8

3

-2

11

12

1

-4

8

10

-2

-6

5

11

6

-8

3

8

-5

-10

5

-12

1

3

-2

-14

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-6$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{3}$
$r=-4$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-3$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=0$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{11}$
$r=1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=2$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$