L11a440

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,3,15,4 X_12,15,5,16 X_8,17,9,18 X_16,7,17,8 X_18,9,19,10 X_22,19,13,20 X_20,12,21,11 X_10,22,11,21 X₂₅₃₆ X_4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 5, -4, 6, -9, 8, -3}, {11, -2, 3, -5, 4, -6, 7, -8, 9, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-uv^{3}w^{2}+uv^{3}w-uv^{2}w^{3}+4uv^{2}w^{2}-3uv^{2}w+uv^{2}+uvw^{3}-3uvw^{2}+4uvw-2uv+uw^{2}-2uw+u+v^{3}\left(-w^{3}\right)+2v^{3}w^{2}-v^{3}w+2v^{2}w^{3}-4v^{2}w^{2}+3v^{2}w-v^{2}-vw^{3}+3vw^{2}-4vw+v-w^{2}+w}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$-q^{-11}+3q^{-10}-6q^{-9}+10q^{-8}-13q^{-7}+17q^{-6}-15q^{-5}+14q^{-4}-10q^{-3}+7q^{-2}-3q^{-1}+1$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-z^{2}a^{10}-2a^{10}+3z^{4}a^{8}+9z^{2}a^{8}+a^{8}z^{-2}+6a^{8}-2z^{6}a^{6}-8z^{4}a^{6}-12z^{2}a^{6}-2a^{6}z^{-2}-9a^{6}-z^{6}a^{4}-z^{4}a^{4}+5z^{2}a^{4}+a^{4}z^{-2}+5a^{4}+z^{4}a^{2}+2z^{2}a^{2}$ (db)
Kauffman polynomial	$a^{13}z^{5}-2a^{13}z^{3}+a^{13}z+3a^{12}z^{6}-6a^{12}z^{4}+3a^{12}z^{2}-a^{12}+4a^{11}z^{7}-5a^{11}z^{5}-a^{11}z^{3}+a^{11}z+4a^{10}z^{8}-3a^{10}z^{6}-3a^{10}z^{4}+2a^{10}z^{2}+3a^{9}z^{9}-7a^{9}z^{5}+9a^{9}z^{3}-3a^{9}z+a^{8}z^{10}+8a^{8}z^{8}-27a^{8}z^{6}+38a^{8}z^{4}-23a^{8}z^{2}-a^{8}z^{-2}+9a^{8}+7a^{7}z^{9}-15a^{7}z^{7}+9a^{7}z^{5}+7a^{7}z^{3}-8a^{7}z+2a^{7}z^{-1}+a^{6}z^{10}+9a^{6}z^{8}-37a^{6}z^{6}+54a^{6}z^{4}-38a^{6}z^{2}-2a^{6}z^{-2}+13a^{6}+4a^{5}z^{9}-8a^{5}z^{7}+2a^{5}z^{5}+3a^{5}z^{3}-5a^{5}z+2a^{5}z^{-1}+5a^{4}z^{8}-15a^{4}z^{6}+16a^{4}z^{4}-14a^{4}z^{2}-a^{4}z^{-2}+6a^{4}+3a^{3}z^{7}-8a^{3}z^{5}+4a^{3}z^{3}+a^{2}z^{6}-3a^{2}z^{4}+2a^{2}z^{2}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

1

-1

2

-2

-3

5

1

4

-5

6

3

-3

-7

8

4

-9

7

6

-1

-11

10

8

2

-13

6

10

4

-15

4

7

-3

-17

2

6

4

-19

1

4

-3

-21

2

-23

1

-1

Integral Khovanov Homology

(db, data source)

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