L11n263

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_14,7,15,8 X_8,13,5,14 X_11,18,12,19 X_22,20,9,19 X_20,16,21,15 X_16,22,17,21 X_17,12,18,13 X₂₅₃₆ X_4,9,1,10
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 $u$ , $v$ , $w$ , ...)	${\frac {-t(1)t(3)^{3}+t(1)t(2)t(3)^{3}-t(2)t(3)^{3}+t(3)^{3}+3t(1)t(3)^{2}-2t(1)t(2)t(3)^{2}+3t(2)t(3)^{2}-3t(3)^{2}-3t(1)t(3)+3t(1)t(2)t(3)-3t(2)t(3)+2t(3)+t(1)-t(1)t(2)+t(2)-1}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{-7}-2q^{-6}+7q^{-5}-8q^{-4}+11q^{-3}-q^{2}-10q^{-2}+4q+9q^{-1}-7$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{8}z^{-2}-2a^{6}z^{-2}-3a^{6}-a^{4}z^{4}+a^{4}z^{2}+a^{4}z^{-2}+3a^{4}+a^{2}z^{6}+3a^{2}z^{4}+3a^{2}z^{2}-z^{4}-z^{2}$ (db)
Kauffman polynomial	$a^{5}z^{9}+a^{3}z^{9}+a^{6}z^{8}+5a^{4}z^{8}+4a^{2}z^{8}-a^{5}z^{7}+5a^{3}z^{7}+6az^{7}-3a^{6}z^{6}-12a^{4}z^{6}-5a^{2}z^{6}+4z^{6}+2a^{7}z^{5}+3a^{5}z^{5}-12a^{3}z^{5}-12az^{5}+z^{5}a^{-1}+a^{8}z^{4}+12a^{6}z^{4}+17a^{4}z^{4}-a^{2}z^{4}-7z^{4}-a^{7}z^{3}+2a^{5}z^{3}+7a^{3}z^{3}+3az^{3}-z^{3}a^{-1}-3a^{8}z^{2}-12a^{6}z^{2}-10a^{4}z^{2}+z^{2}-3a^{7}z-3a^{5}z+3a^{8}+5a^{6}+3a^{4}+2a^{7}z^{-1}+2a^{5}z^{-1}-a^{8}z^{-2}-2a^{6}z^{-2}-a^{4}z^{-2}$ (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

χ

5

1

-1

3

1

4

1

-3

-1

5

3

2

-3

6

5

-1

-5

5

4

1

-7

3

6

3

-9

4

5

-1

-11

1

6

5

-13

1

-1

-15

1

Integral Khovanov Homology

(db, data source)

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