L11n55

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_16,11,17,12 X_7,15,8,14 X_15,9,16,8 X_20,13,21,14 X_22,17,5,18 X_18,21,19,22 X_12,19,13,20 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, 3, -9, 6, 4, -5, -3, 7, -8, 9, -6, 8, -7}

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

Polynomial invariants

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

4

1

3

-4

6

3

-3

-6

6

3

-8

5

6

1

-10

6

0

-12

3

6

3

-14

2

5

-3

-16

3

-18

2

-2

Integral Khovanov Homology

(db, data source)

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