L11a428

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_14,5,15,6 X_20,7,21,8 X_8,19,9,20 X_10,13,5,14 X_16,22,17,21 X_18,12,19,11 X_22,18,11,17 X_2,9,3,10 X_4,16,1,15
Gauss code	{1, -10, 2, -11}, {3, -1, 4, -5, 10, -6}, {8, -2, 6, -3, 11, -7, 9, -8, 5, -4, 7, -9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{2}w^{4}-2uv^{2}w^{3}+2uv^{2}w^{2}-2uv^{2}w+uv^{2}-2uvw^{4}+5uvw^{3}-5uvw^{2}+4uvw-2uv-2uw^{3}+2uw^{2}-2uw+u-v^{2}w^{4}+2v^{2}w^{3}-2v^{2}w^{2}+2v^{2}w+2vw^{4}-4vw^{3}+5vw^{2}-5vw+2v-w^{4}+2w^{3}-2w^{2}+2w-1}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$q^{-6}-q^{5}-3q^{-5}+4q^{4}+8q^{-4}-9q^{3}-12q^{-3}+14q^{2}+19q^{-2}-19q-20q^{-1}+22$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{8}-2a^{2}z^{6}-z^{6}a^{-2}+5z^{6}+a^{4}z^{4}-8a^{2}z^{4}-3z^{4}a^{-2}+10z^{4}+3a^{4}z^{2}-12a^{2}z^{2}-3z^{2}a^{-2}+11z^{2}+3a^{4}-10a^{2}-2a^{-2}+9+2a^{4}z^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}+4z^{-2}$ (db)
Kauffman polynomial	$a^{6}z^{6}-3a^{6}z^{4}+2a^{6}z^{2}+3a^{5}z^{7}-7a^{5}z^{5}+z^{5}a^{-5}+3a^{5}z^{3}-z^{3}a^{-5}+6a^{4}z^{8}-18a^{4}z^{6}+4z^{6}a^{-4}+24a^{4}z^{4}-5z^{4}a^{-4}-22a^{4}z^{2}-2a^{4}z^{-2}+10a^{4}+5a^{3}z^{9}-6a^{3}z^{7}+8z^{7}a^{-3}-9a^{3}z^{5}-13z^{5}a^{-3}+20a^{3}z^{3}+6z^{3}a^{-3}-16a^{3}z-2za^{-3}+5a^{3}z^{-1}+a^{-3}z^{-1}+2a^{2}z^{10}+10a^{2}z^{8}+10z^{8}a^{-2}-41a^{2}z^{6}-19z^{6}a^{-2}+59a^{2}z^{4}+16z^{4}a^{-2}-46a^{2}z^{2}-8z^{2}a^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}+20a^{2}+2a^{-2}+12az^{9}+7z^{9}a^{-1}-23az^{7}-6z^{7}a^{-1}+5az^{5}-7z^{5}a^{-1}+26az^{3}+16z^{3}a^{-1}-27az-13za^{-1}+9az^{-1}+5a^{-1}z^{-1}+2z^{10}+14z^{8}-45z^{6}+53z^{4}-30z^{2}-4z^{-2}+13$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

11

1

-1

9

3

7

6

1

-5

5

8

3

5

3

11

6

-5

1

11

8

3

-1

11

13

2

-3

8

9

-1

-5

5

12

7

-7

3

7

-4

-9

5

-11

1

3

-2

-13

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-1$	$i=1$
$r=-6$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{3}$
$r=-4$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-3$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-2$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{8}$
$r=-1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=0$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{11}$
$r=1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=2$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$