L11n77

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {uv^{5}-4uv^{4}+7uv^{3}-3uv^{2}-3v^{3}+7v^{2}-4v+1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-q^{5/2}+3q^{3/2}-6{\sqrt {q}}+{\frac {8}{\sqrt {q}}}-{\frac {10}{q^{3/2}}}+{\frac {10}{q^{5/2}}}-{\frac {9}{q^{7/2}}}+{\frac {7}{q^{9/2}}}-{\frac {5}{q^{11/2}}}+{\frac {1}{q^{13/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-za^{7}+z^{5}a^{5}+4z^{3}a^{5}+5za^{5}+2a^{5}z^{-1}-z^{7}a^{3}-5z^{5}a^{3}-10z^{3}a^{3}-11za^{3}-4a^{3}z^{-1}+2z^{5}a+7z^{3}a+7za+3az^{-1}-z^{3}a^{-1}-2za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{2}+5a^{7}z^{3}-a^{7}z+3a^{6}z^{6}-2a^{6}z^{4}+a^{6}+8a^{5}z^{7}-26a^{5}z^{5}+31a^{5}z^{3}-14a^{5}z+2a^{5}z^{-1}+7a^{4}z^{8}-22a^{4}z^{6}+22a^{4}z^{4}-12a^{4}z^{2}+2a^{4}+2a^{3}z^{9}+6a^{3}z^{7}-42a^{3}z^{5}+53a^{3}z^{3}-26a^{3}z+4a^{3}z^{-1}+10a^{2}z^{8}-37a^{2}z^{6}+38a^{2}z^{4}-16a^{2}z^{2}+3a^{2}+2az^{9}-az^{7}+z^{7}a^{-1}-20az^{5}-4z^{5}a^{-1}+33az^{3}+6z^{3}a^{-1}-17az-4za^{-1}+3az^{-1}+a^{-1}z^{-1}+3z^{8}-12z^{6}+14z^{4}-5z^{2}+1$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

χ

6

1

4

2

-2

2

4

1

3

0

4

2

-2

6

4

2

-4

5

0

-6

4

5

-1

-8

3

5

2

-10

2

4

-2

-12

4

-14

1

-1

Integral Khovanov Homology

(db, data source)

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