L11n248

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Link Presentations

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Planar diagram presentation	X_12,1,13,2 X₈₄₉₃ X_14,6,15,5 X_18,8,19,7 X_20,9,21,10 X_10,11,1,12 X_6,14,7,13 X_4,18,5,17 X_15,11,16,22 X_2,19,3,20 X_21,17,22,16
Gauss code	{1, -10, 2, -8, 3, -7, 4, -2, 5, -6}, {6, -1, 7, -3, -9, 11, 8, -4, 10, -5, -11, 9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)\left(u^{2}v^{2}-u^{2}v+u^{2}-uv^{2}+3uv-u+v^{2}-v+1\right)}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-6q^{9/2}+10q^{7/2}-{\frac {1}{q^{7/2}}}-14q^{5/2}+{\frac {4}{q^{5/2}}}+15q^{3/2}-{\frac {9}{q^{3/2}}}+2q^{11/2}-15{\sqrt {q}}+{\frac {12}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$za^{-5}+z^{5}a^{-3}+z^{3}a^{-3}-z^{7}a^{-1}+az^{5}-4z^{5}a^{-1}+2az^{3}-6z^{3}a^{-1}+2az-3za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$3z^{4}a^{-6}-3z^{2}a^{-6}+z^{7}a^{-5}+5z^{5}a^{-5}-7z^{3}a^{-5}+2za^{-5}+3z^{8}a^{-4}-z^{6}a^{-4}+z^{4}a^{-4}-z^{2}a^{-4}+2z^{9}a^{-3}+4z^{7}a^{-3}+a^{3}z^{5}-7z^{5}a^{-3}-a^{3}z^{3}+2z^{3}a^{-3}+10z^{8}a^{-2}+4a^{2}z^{6}-14z^{6}a^{-2}-5a^{2}z^{4}+3z^{4}a^{-2}+a^{2}z^{2}+2z^{2}a^{-2}+2z^{9}a^{-1}+8az^{7}+11z^{7}a^{-1}-15az^{5}-28z^{5}a^{-1}+9az^{3}+19z^{3}a^{-1}-4az-6za^{-1}+az^{-1}+a^{-1}z^{-1}+7z^{8}-9z^{6}+z^{2}-1$ (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

12

2

-2

10

4

8

6

2

-4

6

8

4

7

6

-1

2

8

0

6

9

3

-2

3

6

-3

-4

1

6

5

-6

3

-3

-8

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-4$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=0$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{8}$
$r=1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=2$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=4$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=5$	${\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$