L11n279

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₃₈₄₉ X_2,14,3,13 X_14,7,15,8 X_18,10,19,9 X_17,11,18,22 X_11,21,12,20 X_21,17,22,16 X_15,1,16,4 X_10,20,5,19
Gauss code	{1, -4, -3, 10}, {-2, -1, 5, 3, 6, -11}, {-8, 2, 4, -5, -10, 9, -7, -6, 11, 8, -9, 7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2uvw^{2}-2uvw+uv-uw^{2}+uw+v^{2}(-w)+v^{2}-vw^{2}+2vw-2v}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$-2q^{6}+3q^{5}-4q^{4}+6q^{3}-4q^{2}+5q-2+2q^{-1}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-2z^{4}a^{-2}-7z^{2}a^{-2}+4z^{2}a^{-4}+2z^{2}-10a^{-2}+8a^{-4}-2a^{-6}+4-5a^{-2}z^{-2}+4a^{-4}z^{-2}-a^{-6}z^{-2}+2z^{-2}$ (db)
Kauffman polynomial	$3z^{3}a^{-7}-5za^{-7}+a^{-7}z^{-1}+z^{6}a^{-6}-2z^{2}a^{-6}-a^{-6}z^{-2}+3a^{-6}+2z^{7}a^{-5}-8z^{5}a^{-5}+20z^{3}a^{-5}-18za^{-5}+5a^{-5}z^{-1}+z^{8}a^{-4}-3z^{6}a^{-4}+10z^{4}a^{-4}-14z^{2}a^{-4}-4a^{-4}z^{-2}+12a^{-4}+3z^{7}a^{-3}-11z^{5}a^{-3}+24z^{3}a^{-3}-24za^{-3}+9a^{-3}z^{-1}+z^{8}a^{-2}-4z^{6}a^{-2}+13z^{4}a^{-2}-21z^{2}a^{-2}-5a^{-2}z^{-2}+15a^{-2}+z^{7}a^{-1}-3z^{5}a^{-1}+7z^{3}a^{-1}-11za^{-1}+5a^{-1}z^{-1}+3z^{4}-9z^{2}-2z^{-2}+7$ (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		\

-2

-1

0

1

2

3

4

5

χ

13

2

-2

11

1

9

3

2

-1

7

3

1

2

5

2

4

2

3

2

1

4

3

-1

1

0

-3

2

Integral Khovanov Homology

(db, data source)

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