L11n356

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_11,18,12,19 X_16,8,17,7 X_8,16,9,15 X_13,21,14,20 X_19,22,20,15 X_21,13,22,12 X_17,14,18,5 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {5, -4, -9, 3, -7, 6, -8, 7}, {10, -1, 4, -5, 11, -2, -3, 8, -6, 9}

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

Polynomial invariants

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

-5

-4

-3

-2

-1

0

1

2

3

4

χ

7

1

5

2

-2

3

4

1

3

1

4

3

-1

6

3

-3

4

5

1

-5

5

0

-7

2

5

3

-9

2

4

-2

-11

3

-13

1

-1

Integral Khovanov Homology

(db, data source)

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