L11n349

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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_22,18,13,17 X_9,17,10,16 X_15,11,16,10 X_12,20,5,19 X_8,22,9,21 X₂₅₃₆ X_4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -9, -6, 7, 3, -8}, {11, -2, -7, 6, 5, -4, 8, -3, 9, -5}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{2}w^{3}-2uv^{2}w^{2}+2uv^{2}w-uv^{2}-uvw^{3}+2uvw^{2}-3uvw+2uv+uw+v^{3}\left(-w^{2}\right)-2v^{2}w^{3}+3v^{2}w^{2}-2v^{2}w+v^{2}+vw^{3}-2vw^{2}+2vw-v}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$-q^{6}+5q^{5}-7q^{4}+10q^{3}+q^{-3}-10q^{2}-2q^{-2}+10q+6q^{-1}-8$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+2z^{4}a^{-2}-z^{4}a^{-4}-2z^{4}+a^{2}z^{2}-2z^{2}a^{-2}-4z^{2}+2a^{2}-5a^{-2}+3a^{-4}-2a^{-2}z^{-2}+a^{-4}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$z^{3}a^{-7}+5z^{4}a^{-6}-z^{2}a^{-6}-2a^{-6}+2z^{7}a^{-5}+z^{3}a^{-5}+za^{-5}+3z^{8}a^{-4}-5z^{6}a^{-4}+8z^{4}a^{-4}-6z^{2}a^{-4}-a^{-4}z^{-2}+3a^{-4}+z^{9}a^{-3}+5z^{7}a^{-3}-15z^{5}a^{-3}+15z^{3}a^{-3}-8za^{-3}+2a^{-3}z^{-1}+6z^{8}a^{-2}+a^{2}z^{6}-14z^{6}a^{-2}-4a^{2}z^{4}+14z^{4}a^{-2}+5a^{2}z^{2}-16z^{2}a^{-2}-2a^{-2}z^{-2}-2a^{2}+9a^{-2}+z^{9}a^{-1}+2az^{7}+5z^{7}a^{-1}-5az^{5}-20z^{5}a^{-1}+2az^{3}+17z^{3}a^{-1}+az-8za^{-1}+2a^{-1}z^{-1}+3z^{8}-8z^{6}+7z^{4}-6z^{2}-z^{-2}+3$ (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

1

-1

11

4

9

5

3

-2

7

5

2

3

5

0

3

5

0

1

4

6

2

-1

2

4

-2

-3

4

-5

1

2

-1

-7

1

Integral Khovanov Homology

(db, data source)

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