L11n78

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_18,9,19,10 X_8,17,9,18 X_19,1,20,4 X_5,12,6,13 X_3,10,4,11 X_13,22,14,5 X_21,14,22,15 X_11,20,12,21 X_2,16,3,15
Gauss code	{1, -11, -7, 5}, {-6, -1, 2, -4, 3, 7, -10, 6, -8, 9, 11, -2, 4, -3, -5, 10, -9, 8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2(t(1)-1)(t(2)-1)\left(t(2)^{2}-t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {2}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {6}{q^{7/2}}}+{\frac {7}{q^{9/2}}}-{\frac {9}{q^{11/2}}}+{\frac {7}{q^{13/2}}}-{\frac {6}{q^{15/2}}}+{\frac {5}{q^{17/2}}}-{\frac {2}{q^{19/2}}}+{\frac {1}{q^{21/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-z^{3}a^{9}-2za^{9}-2a^{9}z^{-1}+z^{5}a^{7}+3z^{3}a^{7}+5za^{7}+4a^{7}z^{-1}+z^{5}a^{5}+2z^{3}a^{5}+za^{5}-a^{5}z^{-1}-2z^{3}a^{3}-4za^{3}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$-z^{6}a^{12}+4z^{4}a^{12}-5z^{2}a^{12}+2a^{12}-2z^{7}a^{11}+6z^{5}a^{11}-3z^{3}a^{11}-za^{11}-2z^{8}a^{10}+4z^{6}a^{10}+2z^{4}a^{10}-3z^{2}a^{10}+a^{10}-z^{9}a^{9}+3z^{5}a^{9}+4z^{3}a^{9}-6za^{9}+2a^{9}z^{-1}-4z^{8}a^{8}+11z^{6}a^{8}-14z^{4}a^{8}+15z^{2}a^{8}-6a^{8}-z^{9}a^{7}+z^{5}a^{7}+3z^{3}a^{7}-9za^{7}+4a^{7}z^{-1}-2z^{8}a^{6}+5z^{6}a^{6}-12z^{4}a^{6}+12z^{2}a^{6}-5a^{6}-2z^{7}a^{5}+4z^{5}a^{5}-7z^{3}a^{5}+za^{5}+a^{5}z^{-1}-z^{6}a^{4}-z^{2}a^{4}+a^{4}-3z^{3}a^{3}+5za^{3}-a^{3}z^{-1}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-2

2

-4

3

2

-1

-6

3

-8

4

3

-1

-10

5

3

2

-12

2

4

2

-14

4

5

-1

-16

1

2

1

-18

1

4

-3

-20

1

-22

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$
$r=-9$	${\mathbb {Z} }$
$r=-8$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-5$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-4$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-1$	${\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=0$	${\mathbb {Z} }^{2}$	${\mathbb {Z} }^{2}$