L11n177

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_20,9,21,10 X_5,15,6,14 X_18,12,19,11 X_10,4,11,3 X_7,13,8,12 X_13,17,14,16 X_17,7,18,22 X_15,1,16,6 X_4,21,5,22 X_2,20,3,19
Gauss code	{1, -11, 5, -10, -3, 9}, {-6, -1, 2, -5, 4, 6, -7, 3, -9, 7, -8, -4, 11, -2, 10, 8}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{2}v^{4}-2u^{2}v^{3}+3u^{2}v^{2}-3u^{2}v+u^{2}-2uv^{4}+5uv^{3}-5uv^{2}+5uv-2u+v^{4}-3v^{3}+3v^{2}-2v+1}{uv^{2}}}$ (db)
Jones polynomial	$-2q^{15/2}+6q^{13/2}-9q^{11/2}+12q^{9/2}-14q^{7/2}+12q^{5/2}-11q^{3/2}+7{\sqrt {q}}-{\frac {4}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-za^{-7}-z^{5}a^{-5}-z^{3}a^{-5}+za^{-5}+z^{7}a^{-3}+4z^{5}a^{-3}+5z^{3}a^{-3}+za^{-3}-a^{-3}z^{-1}-z^{5}a^{-1}-2z^{3}a^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$3z^{3}a^{-9}-za^{-9}+z^{6}a^{-8}+6z^{4}a^{-8}-5z^{2}a^{-8}+4z^{7}a^{-7}-3z^{5}a^{-7}+4z^{3}a^{-7}-za^{-7}+5z^{8}a^{-6}-8z^{6}a^{-6}+10z^{4}a^{-6}-6z^{2}a^{-6}+2z^{9}a^{-5}+6z^{7}a^{-5}-19z^{5}a^{-5}+12z^{3}a^{-5}-za^{-5}+10z^{8}a^{-4}-21z^{6}a^{-4}+8z^{4}a^{-4}+2z^{9}a^{-3}+6z^{7}a^{-3}-27z^{5}a^{-3}+19z^{3}a^{-3}-2za^{-3}-a^{-3}z^{-1}+5z^{8}a^{-2}-11z^{6}a^{-2}+2z^{4}a^{-2}+2z^{2}a^{-2}+a^{-2}+4z^{7}a^{-1}-11z^{5}a^{-1}+8z^{3}a^{-1}-za^{-1}-a^{-1}z^{-1}+z^{6}-2z^{4}+z^{2}$ (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		\

-3

-2

-1

0

1

2

3

4

5

6

χ

16

2

14

4

-4

12

5

2

3

10

7

4

-3

8

7

5

2

6

8

2

4

5

6

-1

2

3

7

4

0

1

4

-3

-2

3

-4

1

-1

Integral Khovanov Homology

(db, data source)

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