L11n260

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_3,11,4,10 X_7,15,8,14 X_13,5,14,8 X_18,12,19,11 X_19,22,20,9 X_15,20,16,21 X_21,16,22,17 X_12,18,13,17 X₂₅₃₆ X_9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 4}, {-11, 2, 5, -9, -4, 3, -7, 8, 9, -5, -6, 7, -8, 6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {-t(1)t(3)^{3}+t(1)t(2)t(3)^{3}-t(2)t(3)^{3}+t(3)^{3}+2t(1)t(3)^{2}-2t(1)t(2)t(3)^{2}+2t(2)t(3)^{2}-3t(3)^{2}-2t(1)t(3)+3t(1)t(2)t(3)-2t(2)t(3)+2t(3)+t(1)-t(1)t(2)+t(2)-1}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$2q^{4}-4q^{3}+8q^{2}-8q+10-8q^{-1}+7q^{-2}-4q^{-3}+q^{-4}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{-4}z^{-2}+a^{-4}+a^{2}z^{4}+z^{4}a^{-2}+a^{2}z^{2}-2a^{-2}z^{-2}-3a^{-2}-z^{6}-3z^{4}-2z^{2}+z^{-2}+2$ (db)
Kauffman polynomial	$a^{4}z^{4}+3z^{4}a^{-4}-5z^{2}a^{-4}-a^{-4}z^{-2}+3a^{-4}+z^{7}a^{-3}+4a^{3}z^{5}+z^{5}a^{-3}-3a^{3}z^{3}-3za^{-3}+2a^{-3}z^{-1}+z^{8}a^{-2}+7a^{2}z^{6}+3z^{6}a^{-2}-10a^{2}z^{4}-4z^{4}a^{-2}+3a^{2}z^{2}-3z^{2}a^{-2}-2a^{-2}z^{-2}+5a^{-2}+5az^{7}+6z^{7}a^{-1}-3az^{5}-6z^{5}a^{-1}-3az^{3}-3za^{-1}+2a^{-1}z^{-1}+z^{8}+10z^{6}-18z^{4}+5z^{2}-z^{-2}+3$ (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

χ

9

2

7

3

1

-2

5

1

4

3

0

1

7

5

2

-1

4

6

2

-3

3

4

-1

-5

1

4

3

-7

3

-3

-9

1

Integral Khovanov Homology

(db, data source)

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