L11a442

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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_20,12,21,11 X_10,20,11,19 X_12,22,5,21 X₂₅₃₆ X_4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 5, -4, 6, -8, 7, -9}, {11, -2, 3, -5, 4, -6, 8, -7, 9, -3}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{3}w^{3}-uv^{3}w^{2}-uv^{2}w^{3}+2uv^{2}w^{2}-uv^{2}w-uvw^{2}+2uvw-uv-uw+2u-2v^{3}w^{3}+v^{3}w^{2}+v^{2}w^{3}-2v^{2}w^{2}+v^{2}w+vw^{2}-2vw+v+w-1}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$-q^{9}+3q^{8}-4q^{7}+6q^{6}-7q^{5}+8q^{4}-7q^{3}+6q^{2}-4q+4-q^{-1}+q^{-2}$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-6}-4z^{4}a^{-6}-3z^{2}a^{-6}+z^{8}a^{-4}+6z^{6}a^{-4}+12z^{4}a^{-4}+11z^{2}a^{-4}+a^{-4}z^{-2}+5a^{-4}-2z^{6}a^{-2}-11z^{4}a^{-2}-18z^{2}a^{-2}-2a^{-2}z^{-2}-11a^{-2}+z^{4}+5z^{2}+z^{-2}+6$ (db)
Kauffman polynomial	$z^{10}a^{-2}+z^{10}a^{-4}+z^{9}a^{-1}+4z^{9}a^{-3}+3z^{9}a^{-5}-3z^{8}a^{-2}+4z^{8}a^{-6}+z^{8}-4z^{7}a^{-1}-18z^{7}a^{-3}-10z^{7}a^{-5}+4z^{7}a^{-7}-5z^{6}a^{-2}-13z^{6}a^{-4}-11z^{6}a^{-6}+4z^{6}a^{-8}-7z^{6}+18z^{5}a^{-3}+8z^{5}a^{-5}-6z^{5}a^{-7}+4z^{5}a^{-9}+22z^{4}a^{-2}+19z^{4}a^{-4}+8z^{4}a^{-6}-3z^{4}a^{-8}+3z^{4}a^{-10}+17z^{4}+12z^{3}a^{-1}+6z^{3}a^{-3}-2z^{3}a^{-5}-3z^{3}a^{-9}+z^{3}a^{-11}-25z^{2}a^{-2}-10z^{2}a^{-4}-3z^{2}a^{-6}-2z^{2}a^{-8}-2z^{2}a^{-10}-18z^{2}-11za^{-1}-11za^{-3}+13a^{-2}+5a^{-4}+a^{-8}+8+2a^{-1}z^{-1}+2a^{-3}z^{-1}-2a^{-2}z^{-2}-a^{-4}z^{-2}-z^{-2}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

19

1

-1

17

2

15

2

1

-1

13

4

2

11

5

4

-1

9

3

2

1

7

4

5

1

5

2

3

-1

3

5

2

1

0

-1

3

-3

1

0

-5

1

Integral Khovanov Homology

(db, data source)

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