L11a417

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2uv^{2}w^{2}-2uv^{2}w-2uvw^{2}+4uvw-2uv-2uw+3u-3v^{2}w^{2}+2v^{2}w+2vw^{2}-4vw+2v+2w-2}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$-q^{8}+3q^{7}-5q^{6}+8q^{5}-10q^{4}+11q^{3}+q^{-3}-10q^{2}-q^{-2}+9q+4q^{-1}-5$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+z^{6}a^{-4}+3z^{4}a^{-2}+3z^{4}a^{-4}-z^{4}a^{-6}-2z^{4}+a^{2}z^{2}+3z^{2}a^{-2}+2z^{2}a^{-4}-2z^{2}a^{-6}-7z^{2}+3a^{2}+3a^{-2}-6+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$z^{10}a^{-2}+z^{10}a^{-4}+z^{9}a^{-1}+4z^{9}a^{-3}+3z^{9}a^{-5}-3z^{8}a^{-2}+4z^{8}a^{-6}+z^{8}+az^{7}-z^{7}a^{-1}-15z^{7}a^{-3}-9z^{7}a^{-5}+4z^{7}a^{-7}+a^{2}z^{6}+8z^{6}a^{-2}-4z^{6}a^{-4}-10z^{6}a^{-6}+3z^{6}a^{-8}-2az^{5}+26z^{5}a^{-3}+14z^{5}a^{-5}-9z^{5}a^{-7}+z^{5}a^{-9}-5a^{2}z^{4}-10z^{4}a^{-2}+10z^{4}a^{-4}+9z^{4}a^{-6}-7z^{4}a^{-8}-9z^{4}-3az^{3}-5z^{3}a^{-1}-14z^{3}a^{-3}-6z^{3}a^{-5}+4z^{3}a^{-7}-2z^{3}a^{-9}+8a^{2}z^{2}+5z^{2}a^{-2}-4z^{2}a^{-4}-2z^{2}a^{-6}+2z^{2}a^{-8}+13z^{2}+6az+6za^{-1}-5a^{2}-3a^{-2}+a^{-4}-8-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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

17

1

-1

15

2

13

3

1

-2

11

5

2

3

9

5

3

-2

7

6

5

1

5

6

7

1

3

4

-1

1

3

7

4

-1

1

2

-1

-3

3

-5

1

0

-7

1

Integral Khovanov Homology

(db, data source)

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