L11n118

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-2)(v-1)(2v-1)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-10q^{9/2}+11q^{7/2}-13q^{5/2}+11q^{3/2}-{\frac {2}{q^{3/2}}}+q^{15/2}-3q^{13/2}+7q^{11/2}-9{\sqrt {q}}+{\frac {5}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$2z^{5}a^{-3}-4z^{3}a^{-1}+6z^{3}a^{-3}-3z^{3}a^{-5}+2az-6za^{-1}+8za^{-3}-5za^{-5}+za^{-7}+az^{-1}-2a^{-1}z^{-1}+3a^{-3}z^{-1}-3a^{-5}z^{-1}+a^{-7}z^{-1}$ (db)
Kauffman polynomial	$-2z^{9}a^{-3}-2z^{9}a^{-5}-6z^{8}a^{-2}-10z^{8}a^{-4}-4z^{8}a^{-6}-5z^{7}a^{-1}-6z^{7}a^{-3}-4z^{7}a^{-5}-3z^{7}a^{-7}+17z^{6}a^{-2}+27z^{6}a^{-4}+8z^{6}a^{-6}-z^{6}a^{-8}-z^{6}+13z^{5}a^{-1}+30z^{5}a^{-3}+25z^{5}a^{-5}+8z^{5}a^{-7}-24z^{4}a^{-2}-22z^{4}a^{-4}+z^{4}a^{-6}+3z^{4}a^{-8}-4z^{4}-3az^{3}-23z^{3}a^{-1}-39z^{3}a^{-3}-25z^{3}a^{-5}-6z^{3}a^{-7}+11z^{2}a^{-2}+7z^{2}a^{-4}-3z^{2}a^{-6}-3z^{2}a^{-8}+4z^{2}+4az+13za^{-1}+18za^{-3}+12za^{-5}+3za^{-7}-2a^{-2}+2a^{-6}+a^{-8}-az^{-1}-2a^{-1}z^{-1}-3a^{-3}z^{-1}-3a^{-5}z^{-1}-a^{-7}z^{-1}$ (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		\

-2

-1

0

1

2

3

4

5

6

7

χ

16

1

-1

14

2

12

5

1

-4

10

5

2

3

8

6

5

-1

6

7

5

2

4

6

2

5

7

-2

0

2

6

4

-2

3

-3

-4

2

Integral Khovanov Homology

(db, data source)

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