L11n319

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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_11,21,12,20 X_17,11,18,22 X_21,17,22,16 X_10,13,5,14 X_8,20,9,19 X_18,8,19,7 X_2,9,3,10 X_4,16,1,15
Gauss code	{1, -10, 2, -11}, {3, -1, 9, -8, 10, -7}, {-4, -2, 7, -3, 11, 6, -5, -9, 8, 4, -6, 5}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {uv^{2}w^{4}-uv^{2}w^{3}-uvw^{4}+3uvw^{3}-2uvw^{2}+2uvw-uv-uw^{3}+uw^{2}-uw+u-v^{2}w^{4}+v^{2}w^{3}-v^{2}w^{2}+v^{2}w+vw^{4}-2vw^{3}+2vw^{2}-3vw+v+w-1}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$-2q^{6}+4q^{5}-7q^{4}+10q^{3}-9q^{2}+11q-7+6q^{-1}-3q^{-2}+q^{-3}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{8}a^{-2}-6z^{6}a^{-2}+z^{6}a^{-4}+z^{6}-13z^{4}a^{-2}+5z^{4}a^{-4}+4z^{4}-14z^{2}a^{-2}+9z^{2}a^{-4}-z^{2}a^{-6}+5z^{2}-10a^{-2}+8a^{-4}-2a^{-6}+4-5a^{-2}z^{-2}+4a^{-4}z^{-2}-a^{-6}z^{-2}+2z^{-2}$ (db)
Kauffman polynomial	$2z^{9}a^{-1}+2z^{9}a^{-3}+9z^{8}a^{-2}+5z^{8}a^{-4}+4z^{8}+3az^{7}-z^{7}a^{-1}+4z^{7}a^{-5}+a^{2}z^{6}-34z^{6}a^{-2}-19z^{6}a^{-4}+z^{6}a^{-6}-13z^{6}-9az^{5}-11z^{5}a^{-1}-15z^{5}a^{-3}-13z^{5}a^{-5}-3a^{2}z^{4}+50z^{4}a^{-2}+35z^{4}a^{-4}+2z^{4}a^{-6}+14z^{4}+5az^{3}+14z^{3}a^{-1}+31z^{3}a^{-3}+25z^{3}a^{-5}+3z^{3}a^{-7}+2a^{2}z^{2}-37z^{2}a^{-2}-26z^{2}a^{-4}-4z^{2}a^{-6}-13z^{2}-11za^{-1}-24za^{-3}-18za^{-5}-5za^{-7}+15a^{-2}+12a^{-4}+3a^{-6}+7+5a^{-1}z^{-1}+9a^{-3}z^{-1}+5a^{-5}z^{-1}+a^{-7}z^{-1}-5a^{-2}z^{-2}-4a^{-4}z^{-2}-a^{-6}z^{-2}-2z^{-2}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

χ

13

2

-2

11

2

9

5

2

-3

7

5

2

3

5

6

1

3

6

4

2

1

3

7

4

-1

3

4

-1

-3

1

4

3

-5

2

-2

-7

1

Integral Khovanov Homology

(db, data source)

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