L11n234

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_5,20,6,21 X_3,15,4,14 X_15,5,16,4 X_19,22,20,9 X_7,17,8,16 X_11,18,12,19 X_17,12,18,13 X_2,9,3,10 X_13,1,14,8 X_21,6,22,7
Gauss code	{1, -9, -3, 4, -2, 11, -6, 10}, {9, -1, -7, 8, -10, 3, -4, 6, -8, 7, -5, 2, -11, 5}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(1)^{2}t(2)^{2}-t(1)t(2)^{2}+5t(1)t(2)-t(1)+1\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {5}{q^{9/2}}}-q^{7/2}+{\frac {8}{q^{7/2}}}+3q^{5/2}-{\frac {12}{q^{5/2}}}-7q^{3/2}+{\frac {12}{q^{3/2}}}+{\frac {2}{q^{11/2}}}+10{\sqrt {q}}-{\frac {12}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{5}z-a^{3}z^{5}-2a^{3}z^{3}-2a^{3}z+az^{7}+5az^{5}-z^{5}a^{-1}+10az^{3}-3z^{3}a^{-1}+7az-4za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{9}-az^{9}-2a^{4}z^{8}-6a^{2}z^{8}-4z^{8}-a^{5}z^{7}-5a^{3}z^{7}-9az^{7}-5z^{7}a^{-1}+6a^{2}z^{6}-3z^{6}a^{-2}+3z^{6}-3a^{5}z^{5}+9a^{3}z^{5}+22az^{5}+9z^{5}a^{-1}-z^{5}a^{-3}-3a^{6}z^{4}+a^{2}z^{4}+5z^{4}a^{-2}+3z^{4}+6a^{5}z^{3}-5a^{3}z^{3}-19az^{3}-6z^{3}a^{-1}+2z^{3}a^{-3}+4a^{6}z^{2}+a^{4}z^{2}-2a^{2}z^{2}-2z^{2}a^{-2}-z^{2}-3a^{5}z+8az+4za^{-1}-za^{-3}+1-az^{-1}-a^{-1}z^{-1}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

2

-2

4

5

1

4

2

5

2

-3

0

7

5

2

-2

7

0

-4

5

0

-6

3

7

4

-8

2

5

-3

-10

3

-12

2

-2

Integral Khovanov Homology

(db, data source)

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