L11n296

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_15,20,16,21 X_14,8,15,7 X_10,22,5,21 X_18,11,19,12 X_9,17,10,16 X_22,17,11,18 X_19,9,20,8 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 4, 9, -7, -5}, {6, -2, 11, -4, -3, 7, 8, -6, -9, 3, 5, -8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {uv^{2}w+uvw^{4}-uvw^{3}+uvw^{2}-2uvw+uv-uw^{4}+uw^{3}-v^{2}w+v^{2}-vw^{4}+2vw^{3}-vw^{2}+vw-v-w^{3}}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$q^{6}-2q^{5}+4q^{4}-4q^{3}+7q^{2}-5q+6-4q^{-1}+2q^{-2}-q^{-3}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{4}a^{-4}+3z^{2}a^{-4}+2a^{-4}z^{-2}+3a^{-4}-z^{6}a^{-2}-5z^{4}a^{-2}-a^{2}z^{2}-10z^{2}a^{-2}-a^{2}z^{-2}-5a^{-2}z^{-2}-2a^{2}-10a^{-2}+2z^{4}+7z^{2}+4z^{-2}+9$ (db)
Kauffman polynomial	$z^{9}a^{-1}+z^{9}a^{-3}+5z^{8}a^{-2}+3z^{8}a^{-4}+2z^{8}+az^{7}-z^{7}a^{-1}+2z^{7}a^{-5}-25z^{6}a^{-2}-14z^{6}a^{-4}+z^{6}a^{-6}-10z^{6}-4az^{5}-10z^{5}a^{-1}-13z^{5}a^{-3}-7z^{5}a^{-5}+2a^{2}z^{4}+48z^{4}a^{-2}+23z^{4}a^{-4}-4z^{4}a^{-6}+23z^{4}+a^{3}z^{3}+12az^{3}+30z^{3}a^{-1}+23z^{3}a^{-3}+4z^{3}a^{-5}-3a^{2}z^{2}-44z^{2}a^{-2}-22z^{2}a^{-4}+3z^{2}a^{-6}-22z^{2}-2a^{3}z-13az-27za^{-1}-16za^{-3}+2a^{2}+20a^{-2}+10a^{-4}+13+a^{3}z^{-1}+5az^{-1}+9a^{-1}z^{-1}+5a^{-3}z^{-1}-a^{2}z^{-2}-5a^{-2}z^{-2}-2a^{-4}z^{-2}-4z^{-2}$ (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		\

-3

-2

-1

0

1

2

3

4

5

6

χ

13

1

11

2

1

-1

9

2

7

2

0

5

2

3

1

3

2

1

5

4

1

-1

1

3

2

-3

1

3

-2

-5

1

-7

1

-1

Integral Khovanov Homology

(db, data source)

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