L11a499

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_16,8,17,7 X_22,17,19,18 X_20,12,21,11 X_10,20,11,19 X_18,21,5,22 X_14,10,15,9 X_8,16,9,15 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -10, 2, -11}, {6, -5, 7, -4}, {10, -1, 3, -9, 8, -6, 5, -2, 11, -8, 9, -3, 4, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)(t(3)-1)\left(2t(3)^{2}-3t(3)+2\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{8}+3q^{7}-7q^{6}+12q^{5}-16q^{4}+19q^{3}-17q^{2}+16q-11+7q^{-1}-2q^{-2}+q^{-3}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+z^{6}a^{-4}+z^{4}a^{-2}+3z^{4}a^{-4}-z^{4}a^{-6}-2z^{4}+a^{2}z^{2}-z^{2}a^{-2}+6z^{2}a^{-4}-2z^{2}a^{-6}-4z^{2}+2a^{2}+4a^{-4}-2a^{-6}-4+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$z^{5}a^{-9}-2z^{3}a^{-9}+3z^{6}a^{-8}-5z^{4}a^{-8}+6z^{7}a^{-7}-13z^{5}a^{-7}+11z^{3}a^{-7}-4za^{-7}+8z^{8}a^{-6}-21z^{6}a^{-6}+27z^{4}a^{-6}-14z^{2}a^{-6}+4a^{-6}+5z^{9}a^{-5}-4z^{7}a^{-5}-9z^{5}a^{-5}+21z^{3}a^{-5}-8za^{-5}+z^{10}a^{-4}+14z^{8}a^{-4}-46z^{6}a^{-4}+62z^{4}a^{-4}-36z^{2}a^{-4}+8a^{-4}+8z^{9}a^{-3}-14z^{7}a^{-3}+9z^{5}a^{-3}-3z^{3}a^{-3}+z^{10}a^{-2}+9z^{8}a^{-2}+a^{2}z^{6}-26z^{6}a^{-2}-4a^{2}z^{4}+27z^{4}a^{-2}+6a^{2}z^{2}-16z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-4a^{2}+3z^{9}a^{-1}+2az^{7}-2z^{7}a^{-1}-4az^{5}-11z^{3}a^{-1}+4az+8za^{-1}-2az^{-1}-2a^{-1}z^{-1}+3z^{8}-3z^{6}-7z^{4}+12z^{2}+2z^{-2}-7$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

17

1

-1

15

2

13

5

1

-4

11

7

2

5

9

5

-4

7

10

7

3

5

7

9

2

3

9

10

-1

1

6

11

5

-1

1

5

-4

-3

1

6

5

-5

1

-1

-7

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=1$	$i=3$
$r=-4$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=0$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{9}$
$r=1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=3$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=4$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=5$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=6$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=7$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$