L11n340

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_18,12,19,11 X_7,14,8,15 X_13,8,14,9 X_22,20,13,19 X_20,16,21,15 X_16,22,17,21 X_12,18,5,17 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, 3, -9}, {-5, 4, 7, -8, 9, -3, 6, -7, 8, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(v-1)(w-1)\left(-uvw^{2}+2uvw-uv-uw+v^{2}w+vw^{2}-2vw+v\right)}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$-q^{5}+3q^{4}-7q^{3}+11q^{2}-13q+14-12q^{-1}+11q^{-2}-5q^{-3}+3q^{-4}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{2}a^{-4}+2a^{4}z^{-2}+2a^{4}-a^{-4}+a^{2}z^{4}+2z^{4}a^{-2}-2a^{2}z^{2}+3z^{2}a^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}-7a^{2}+a^{-2}-z^{6}-2z^{4}+4z^{-2}+5$ (db)
Kauffman polynomial	$z^{5}a^{-5}-2z^{3}a^{-5}+za^{-5}+3z^{6}a^{-4}+6a^{4}z^{4}-5z^{4}a^{-4}-13a^{4}z^{2}+3z^{2}a^{-4}-2a^{4}z^{-2}+9a^{4}-a^{-4}+3a^{3}z^{7}+5z^{7}a^{-3}-6a^{3}z^{5}-8z^{5}a^{-3}+14a^{3}z^{3}+5z^{3}a^{-3}-16a^{3}z-3za^{-3}+5a^{3}z^{-1}+a^{-3}z^{-1}+4a^{2}z^{8}+4z^{8}a^{-2}-8a^{2}z^{6}+18a^{2}z^{4}-10z^{4}a^{-2}-26a^{2}z^{2}+6z^{2}a^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}+18a^{2}+az^{9}+z^{9}a^{-1}+11az^{7}+13z^{7}a^{-1}-32az^{5}-35z^{5}a^{-1}+43az^{3}+36z^{3}a^{-1}-32az-20za^{-1}+9az^{-1}+5a^{-1}z^{-1}+8z^{8}-11z^{6}+7z^{4}-10z^{2}-4z^{-2}+11$ (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

χ

11

1

-1

9

2

7

5

1

-4

5

6

2

4

3

7

5

-2

1

7

6

1

-1

7

9

2

-3

4

5

-1

-5

1

7

6

-7

2

4

-2

-9

3

Integral Khovanov Homology

(db, data source)

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