L11n57

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_7,14,8,15 X_11,19,12,18 X_19,5,20,22 X_15,21,16,20 X_21,17,22,16 X_17,13,18,12 X_13,8,14,9 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, -4, 8, -9, 3, -6, 7, -8, 4, -5, 6, -7, 5}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(2)^{5}-t(1)t(2)^{4}-3t(2)^{4}+2t(2)^{3}+2t(1)t(2)^{2}-3t(1)t(2)-t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{9/2}+{\frac {2}{q^{9/2}}}-q^{7/2}-{\frac {4}{q^{7/2}}}+{\frac {4}{q^{5/2}}}+q^{3/2}-{\frac {4}{q^{3/2}}}-{\frac {1}{q^{11/2}}}-3{\sqrt {q}}+{\frac {3}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{5}z+a^{5}z^{-1}-2a^{3}z^{3}+z^{3}a^{-3}-4a^{3}z-2a^{3}z^{-1}+3za^{-3}+a^{-3}z^{-1}+az^{5}-z^{5}a^{-1}+4az^{3}-6z^{3}a^{-1}+6az+3az^{-1}-8za^{-1}-3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{9}-az^{9}-2a^{4}z^{8}-4a^{2}z^{8}-2z^{8}-a^{5}z^{7}+a^{3}z^{7}+2az^{7}-z^{7}a^{-1}-z^{7}a^{-3}+9a^{4}z^{6}+19a^{2}z^{6}-z^{6}a^{-4}+11z^{6}+5a^{5}z^{5}+12a^{3}z^{5}+10az^{5}+9z^{5}a^{-1}+6z^{5}a^{-3}-10a^{4}z^{4}-23a^{2}z^{4}+4z^{4}a^{-2}+5z^{4}a^{-4}-14z^{4}-8a^{5}z^{3}-20a^{3}z^{3}-22az^{3}-18z^{3}a^{-1}-8z^{3}a^{-3}+3a^{4}z^{2}+10a^{2}z^{2}-7z^{2}a^{-2}-5z^{2}a^{-4}+5z^{2}+5a^{5}z+10a^{3}z+14az+13za^{-1}+4za^{-3}-2a^{2}+2a^{-2}+a^{-4}-a^{5}z^{-1}-2a^{3}z^{-1}-3az^{-1}-3a^{-1}z^{-1}-a^{-3}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

0

6

1

4

2

1

-1

2

1

2

0

3

4

1

0

-2

2

1

-4

2

3

1

0

-6

2

0

-8

1

3

2

-10

1

-1

-12

1

Integral Khovanov Homology

(db, data source)

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