L11n102

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_3,13,4,12 X_7,16,8,17 X_17,22,18,5 X_13,18,14,19 X_21,14,22,15 X_9,20,10,21 X_15,8,16,9 X_19,10,20,11 X₂₅₃₆ X_11,1,12,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 8, -7, 9, -11, 2, -5, 6, -8, 3, -4, 5, -9, 7, -6, 4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(3t(2)^{2}-4t(2)+3\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {7}{q^{3/2}}}+{\frac {10}{q^{5/2}}}-{\frac {14}{q^{7/2}}}+{\frac {13}{q^{9/2}}}-{\frac {13}{q^{11/2}}}+{\frac {10}{q^{13/2}}}-{\frac {6}{q^{15/2}}}+{\frac {3}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{9}z^{-1}-2a^{7}z^{3}-a^{7}z+a^{7}z^{-1}+2a^{5}z^{5}+5a^{5}z^{3}+5a^{5}z+2a^{5}z^{-1}+a^{3}z^{5}-3a^{3}z-2a^{3}z^{-1}-az^{3}-az$ (db)
Kauffman polynomial	$-6z^{4}a^{10}+11z^{2}a^{10}-4a^{10}-3z^{7}a^{9}+3z^{5}a^{9}-z^{3}a^{9}+2za^{9}+a^{9}z^{-1}-5z^{8}a^{8}+13z^{6}a^{8}-24z^{4}a^{8}+25z^{2}a^{8}-9a^{8}-2z^{9}a^{7}-5z^{7}a^{7}+20z^{5}a^{7}-25z^{3}a^{7}+9za^{7}+a^{7}z^{-1}-10z^{8}a^{6}+23z^{6}a^{6}-24z^{4}a^{6}+12z^{2}a^{6}-4a^{6}-2z^{9}a^{5}-7z^{7}a^{5}+27z^{5}a^{5}-33z^{3}a^{5}+15za^{5}-2a^{5}z^{-1}-5z^{8}a^{4}+7z^{6}a^{4}-z^{4}a^{4}-3z^{2}a^{4}+2a^{4}-5z^{7}a^{3}+9z^{5}a^{3}-7z^{3}a^{3}+7za^{3}-2a^{3}z^{-1}-3z^{6}a^{2}+5z^{4}a^{2}-z^{2}a^{2}-z^{5}a+2z^{3}a-za$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

2

1

0

2

-2

5

1

4

-4

7

4

-3

-6

7

3

4

-8

6

7

1

-10

7

0

-12

3

6

3

-14

3

7

-4

-16

3

-18

3

-3

Integral Khovanov Homology

(db, data source)

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