L11a455

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,3,15,4 X_20,12,21,11 X_18,8,19,7 X_16,10,17,9 X_8,18,9,17 X_22,16,13,15 X_12,20,5,19 X_10,22,11,21 X₂₅₃₆ X_4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -6, 5, -9, 3, -8}, {11, -2, 7, -5, 6, -4, 8, -3, 9, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{3}w-uv^{3}+2uv^{2}w^{2}-4uv^{2}w+2uv^{2}+uvw^{3}-4uvw^{2}+3uvw-uw^{3}+2uw^{2}-2v^{3}w+v^{3}-3v^{2}w^{2}+4v^{2}w-v^{2}-2vw^{3}+4vw^{2}-2vw+w^{3}-w^{2}}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$-q^{7}+3q^{6}-5q^{5}+10q^{4}+q^{-4}-11q^{3}-2q^{-3}+13q^{2}+5q^{-2}-13q-8q^{-1}+12$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{4}-2z^{2}a^{2}+z^{4}-2z^{2}+z^{-2}+2z^{4}a^{-2}-2a^{-2}z^{-2}-3a^{-2}+z^{4}a^{-4}+a^{-4}z^{-2}+2a^{-4}-z^{2}a^{-6}$ (db)
Kauffman polynomial	$z^{10}a^{-2}+z^{10}a^{-4}+3z^{9}a^{-1}+6z^{9}a^{-3}+3z^{9}a^{-5}+7z^{8}a^{-2}+5z^{8}a^{-4}+3z^{8}a^{-6}+5z^{8}+4az^{7}-14z^{7}a^{-3}-9z^{7}a^{-5}+z^{7}a^{-7}+3a^{2}z^{6}-30z^{6}a^{-2}-29z^{6}a^{-4}-13z^{6}a^{-6}-11z^{6}+2a^{3}z^{5}-4az^{5}-16z^{5}a^{-1}-2z^{5}a^{-3}+4z^{5}a^{-5}-4z^{5}a^{-7}+a^{4}z^{4}-2a^{2}z^{4}+39z^{4}a^{-2}+39z^{4}a^{-4}+18z^{4}a^{-6}+15z^{4}-2a^{3}z^{3}+2az^{3}+23z^{3}a^{-1}+17z^{3}a^{-3}+2z^{3}a^{-5}+4z^{3}a^{-7}-2a^{4}z^{2}-29z^{2}a^{-2}-26z^{2}a^{-4}-9z^{2}a^{-6}-10z^{2}-11za^{-1}-11za^{-3}+a^{4}+13a^{-2}+8a^{-4}+5+2a^{-1}z^{-1}+2a^{-3}z^{-1}-2a^{-2}z^{-2}-a^{-4}z^{-2}-z^{-2}$ (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

6

7

χ

15

1

-1

13

2

11

3

1

-2

9

7

2

5

7

6

5

-1

5

7

5

2

3

6

0

1

6

7

-1

3

7

4

-3

2

5

-3

-5

3

-7

1

2

-1

-9

1

Integral Khovanov Homology

(db, data source)

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