L11n251

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Link Presentations

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Planar diagram presentation	X_12,1,13,2 X_2,13,3,14 X₈₃₉₄ X_11,17,12,16 X_14,8,15,7 X_6,16,7,15 X_17,11,18,22 X_4,20,5,19 X_18,6,19,5 X_20,9,21,10 X_10,21,1,22
Gauss code	{1, -2, 3, -8, 9, -6, 5, -3, 10, -11}, {-4, -1, 2, -5, 6, 4, -7, -9, 8, -10, 11, 7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}t(1)^{2}-t(2)t(1)^{2}+t(1)^{2}+3t(2)t(1)+t(2)^{2}-t(2)+1\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {3}{q^{9/2}}}-2q^{7/2}+{\frac {7}{q^{7/2}}}+5q^{5/2}-{\frac {11}{q^{5/2}}}-8q^{3/2}+{\frac {11}{q^{3/2}}}+{\frac {1}{q^{11/2}}}+11{\sqrt {q}}-{\frac {13}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{3}z^{5}-3a^{3}z^{3}-4a^{3}z-za^{-3}+az^{7}+5az^{5}-z^{5}a^{-1}+10az^{3}-2z^{3}a^{-1}+7az-2za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{6}z^{4}-a^{6}z^{2}+3a^{5}z^{5}-2a^{5}z^{3}+6a^{4}z^{6}-7a^{4}z^{4}+4a^{4}z^{2}+8a^{3}z^{7}-15a^{3}z^{5}+3z^{5}a^{-3}+16a^{3}z^{3}-6z^{3}a^{-3}-8a^{3}z+2za^{-3}+5a^{2}z^{8}+z^{8}a^{-2}-3a^{2}z^{6}+2z^{6}a^{-2}-6a^{2}z^{4}-9z^{4}a^{-2}+7a^{2}z^{2}+6z^{2}a^{-2}+az^{9}+z^{9}a^{-1}+11az^{7}+3z^{7}a^{-1}-30az^{5}-9z^{5}a^{-1}+29az^{3}+5z^{3}a^{-1}-14az-4za^{-1}+az^{-1}+a^{-1}z^{-1}+6z^{8}-7z^{6}-7z^{4}+8z^{2}-1$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

2

6

3

-3

4

5

2

3

2

6

3

-3

0

7

5

2

-2

6

8

2

-4

5

0

-6

2

6

4

-8

1

5

-4

-10

2

-12

1

-1

Integral Khovanov Homology

(db, data source)

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