L11a515

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_14,4,15,3 X_20,11,21,12 X_18,10,19,9 X_22,19,13,20 X_10,14,11,13 X_12,21,7,22 X_16,6,17,5 X₂₇₃₈ X_4,16,5,15 X_6,18,1,17
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 4, -6, 3, -7}, {6, -2, 10, -8, 11, -4, 5, -3, 7, -5}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{2}v^{2}w^{3}-u^{2}v^{2}w^{2}-2u^{2}vw^{3}+4u^{2}vw^{2}-u^{2}vw+u^{2}w^{3}-2u^{2}w^{2}+u^{2}w-uv^{2}w^{3}+2uv^{2}w^{2}-uv^{2}w+uvw^{3}-4uvw^{2}+4uvw-uv+uw^{2}-2uw+u-v^{2}w^{2}+2v^{2}w-v^{2}+vw^{2}-4vw+2v+w-1}{uvw^{3/2}}}$ (db)
Jones polynomial	$q^{7}-3q^{6}+6q^{5}-9q^{4}-q^{-4}+13q^{3}+3q^{-3}-13q^{2}-5q^{-2}+14q+9q^{-1}-11$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-4}+4z^{4}a^{-4}+5z^{2}a^{-4}+a^{-4}z^{-2}+3a^{-4}-z^{8}a^{-2}-6z^{6}a^{-2}-a^{2}z^{4}-14z^{4}a^{-2}-3a^{2}z^{2}-16z^{2}a^{-2}-2a^{-2}z^{-2}-a^{2}-8a^{-2}+2z^{6}+9z^{4}+12z^{2}+z^{-2}+6$ (db)
Kauffman polynomial	$z^{4}a^{-8}-z^{2}a^{-8}+3z^{5}a^{-7}-3z^{3}a^{-7}+5z^{6}a^{-6}-6z^{4}a^{-6}+3z^{2}a^{-6}-a^{-6}+6z^{7}a^{-5}-8z^{5}a^{-5}+4z^{3}a^{-5}+6z^{8}a^{-4}-11z^{6}a^{-4}+10z^{4}a^{-4}-6z^{2}a^{-4}-a^{-4}z^{-2}+4a^{-4}+4z^{9}a^{-3}+a^{3}z^{7}-5z^{7}a^{-3}-4a^{3}z^{5}-3z^{5}a^{-3}+4a^{3}z^{3}+8z^{3}a^{-3}-a^{3}z-6za^{-3}+2a^{-3}z^{-1}+z^{10}a^{-2}+3a^{2}z^{8}+9z^{8}a^{-2}-13a^{2}z^{6}-40z^{6}a^{-2}+17a^{2}z^{4}+56z^{4}a^{-2}-9a^{2}z^{2}-37z^{2}a^{-2}-2a^{-2}z^{-2}+2a^{2}+12a^{-2}+3az^{9}+7z^{9}a^{-1}-9az^{7}-21z^{7}a^{-1}+2az^{5}+14z^{5}a^{-1}+7az^{3}+4z^{3}a^{-1}-3az-8za^{-1}+2a^{-1}z^{-1}+z^{10}+6z^{8}-37z^{6}+56z^{4}-36z^{2}-z^{-2}+10$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

15

1

13

2

-2

11

4

1

3

9

6

3

-3

7

3

4

5

7

0

3

7

6

1

5

8

3

-1

4

6

-2

-3

2

6

4

-5

1

3

-2

-7

2

-9

1

-1

Integral Khovanov Homology

(db, data source)

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