L11a443

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Link Presentations

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

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

Polynomial invariants

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

17

1

-1

15

2

13

3

1

-2

11

7

2

5

9

8

5

-3

7

5

2

5

7

8

1

3

6

7

-1

1

4

8

4

-1

2

5

-3

4

-5

1

2

-1

-7

1

Integral Khovanov Homology

(db, data source)

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