L11n452

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_11,20,12,21 X_13,19,14,22 X_21,18,22,9 X_17,13,18,12 X_8,16,5,15 X_14,8,15,7 X_19,17,20,16 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 8, -7}, {-9, 3, -5, 4}, {11, -2, -3, 6, -4, -8, 7, 9, -6, 5}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(3)-1)(t(4)-1)^{2}(t(2)t(4)-t(1))}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}t(4)^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}+{\frac {2}{q^{9/2}}}-q^{7/2}-{\frac {6}{q^{7/2}}}-q^{5/2}+{\frac {5}{q^{5/2}}}+2q^{3/2}-{\frac {7}{q^{3/2}}}-{\frac {1}{q^{11/2}}}-6{\sqrt {q}}+{\frac {4}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{5}z^{-3}+a^{5}z+2a^{5}z^{-1}-3a^{3}z^{3}-3a^{3}z^{-3}+z^{3}a^{-3}-8a^{3}z-8a^{3}z^{-1}+3za^{-3}+a^{-3}z^{-1}+2az^{5}-z^{5}a^{-1}+9az^{3}+3az^{-3}-7z^{3}a^{-1}-a^{-1}z^{-3}+15az+11az^{-1}-11za^{-1}-6a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{5}z^{7}-5a^{5}z^{5}+10a^{5}z^{3}-a^{5}z^{-3}-10a^{5}z+5a^{5}z^{-1}+2a^{4}z^{8}-7a^{4}z^{6}+z^{6}a^{-4}+3a^{4}z^{4}-5z^{4}a^{-4}+7a^{4}z^{2}+4z^{2}a^{-4}+3a^{4}z^{-2}-9a^{4}-a^{-4}+a^{3}z^{9}+3a^{3}z^{7}+z^{7}a^{-3}-29a^{3}z^{5}-7z^{5}a^{-3}+45a^{3}z^{3}+11z^{3}a^{-3}-3a^{3}z^{-3}-34a^{3}z-9za^{-3}+14a^{3}z^{-1}+2a^{-3}z^{-1}+6a^{2}z^{8}-23a^{2}z^{6}+12a^{2}z^{4}-7z^{4}a^{-2}+19a^{2}z^{2}+15z^{2}a^{-2}+6a^{2}z^{-2}-21a^{2}-6a^{-2}+az^{9}+5az^{7}+4z^{7}a^{-1}-44az^{5}-27z^{5}a^{-1}+75az^{3}+51z^{3}a^{-1}-3az^{-3}-a^{-1}z^{-3}-50az-35za^{-1}+18az^{-1}+11a^{-1}z^{-1}+4z^{8}-17z^{6}+7z^{4}+23z^{2}+3z^{-2}-18$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

0

6

2

1

2

4

2

1

-1

2

5

2

1

4

0

5

8

1

2

-2

2

3

4

3

-4

3

5

2

-6

3

2

1

-8

1

5

4

-10

1

-1

-12

1

Integral Khovanov Homology

(db, data source)

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