L11n50

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,4,11,3 X_12,8,13,7 X_13,18,14,19 X_9,17,10,16 X_17,9,18,8 X_19,22,20,5 X_15,21,16,20 X_21,15,22,14 X₂₅₃₆ X_4,12,1,11
Gauss code	{1, -10, 2, -11}, {10, -1, 3, 6, -5, -2, 11, -3, -4, 9, -8, 5, -6, 4, -7, 8, -9, 7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}-4t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-7q^{9/2}+8q^{7/2}-8q^{5/2}+7q^{3/2}-{\frac {1}{q^{3/2}}}+q^{15/2}-3q^{13/2}+5q^{11/2}-6{\sqrt {q}}+{\frac {2}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$za^{-7}-2z^{3}a^{-5}-2za^{-5}+z^{5}a^{-3}+2z^{3}a^{-3}+2za^{-3}-2z^{3}a^{-1}+az-2za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-8}-3z^{4}a^{-8}+2z^{2}a^{-8}+3z^{7}a^{-7}-10z^{5}a^{-7}+8z^{3}a^{-7}-2za^{-7}+3z^{8}a^{-6}-8z^{6}a^{-6}+3z^{4}a^{-6}-z^{2}a^{-6}+z^{9}a^{-5}+3z^{7}a^{-5}-17z^{5}a^{-5}+15z^{3}a^{-5}-4za^{-5}+5z^{8}a^{-4}-15z^{6}a^{-4}+14z^{4}a^{-4}-5z^{2}a^{-4}+z^{9}a^{-3}+z^{7}a^{-3}-9z^{5}a^{-3}+13z^{3}a^{-3}-4za^{-3}+2z^{8}a^{-2}-6z^{6}a^{-2}+10z^{4}a^{-2}-3z^{2}a^{-2}+z^{7}a^{-1}-2z^{5}a^{-1}+az^{3}+7z^{3}a^{-1}-2az-4za^{-1}+az^{-1}+a^{-1}z^{-1}+2z^{4}-z^{2}-1$ (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

6

7

χ

16

1

-1

14

2

12

3

1

-2

10

4

2

8

4

3

-1

6

4

0

4

3

4

1

2

3

4

-1

0

1

5

4

-2

1

-1

-4

1

Integral Khovanov Homology

(db, data source)

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