L11a228

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Link Presentations

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

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

Polynomial invariants

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

8

3

5

8

11

6

-5

6

11

8

3

4

10

11

1

2

9

11

-2

0

5

11

6

-2

3

8

-5

-4

1

6

5

-6

2

-2

-8

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-4$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{3}$
$r=-1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=0$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{9}$
$r=1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=3$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=4$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=6$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=7$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$