L11n70

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(2)^{5}-t(2)^{5}-2t(1)t(2)^{4}+t(2)^{4}+t(1)t(2)^{3}-t(2)^{3}-t(1)t(2)^{2}+t(2)^{2}+t(1)t(2)-2t(2)-t(1)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{\sqrt {q}}}+{\frac {2}{q^{3/2}}}-{\frac {3}{q^{5/2}}}+{\frac {3}{q^{7/2}}}-{\frac {5}{q^{9/2}}}+{\frac {4}{q^{11/2}}}-{\frac {4}{q^{13/2}}}+{\frac {3}{q^{15/2}}}-{\frac {2}{q^{17/2}}}+{\frac {1}{q^{19/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-a^{9}z^{-1}-z^{5}a^{7}-3z^{3}a^{7}+za^{7}+3a^{7}z^{-1}+z^{7}a^{5}+5z^{5}a^{5}+6z^{3}a^{5}-2a^{5}z^{-1}-z^{5}a^{3}-4z^{3}a^{3}-3za^{3}$ (db)
Kauffman polynomial	$-z^{2}a^{12}-2z^{3}a^{11}-3z^{4}a^{10}+3z^{2}a^{10}-a^{10}-4z^{5}a^{9}+8z^{3}a^{9}-4za^{9}+a^{9}z^{-1}-z^{8}a^{8}+3z^{6}a^{8}-6z^{4}a^{8}+10z^{2}a^{8}-3a^{8}-z^{9}a^{7}+4z^{7}a^{7}-8z^{5}a^{7}+14z^{3}a^{7}-10za^{7}+3a^{7}z^{-1}-3z^{8}a^{6}+13z^{6}a^{6}-16z^{4}a^{6}+9z^{2}a^{6}-3a^{6}-z^{9}a^{5}+3z^{7}a^{5}+z^{5}a^{5}-3z^{3}a^{5}-3za^{5}+2a^{5}z^{-1}-2z^{8}a^{4}+10z^{6}a^{4}-13z^{4}a^{4}+3z^{2}a^{4}-z^{7}a^{3}+5z^{5}a^{3}-7z^{3}a^{3}+3za^{3}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

0

1

-2

1

-1

-4

2

1

-6

2

0

-8

3

1

2

-10

1

2

1

-12

3

0

-14

1

2

1

-16

1

2

-1

-18

1

-20

1

-1

Integral Khovanov Homology

(db, data source)

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