L11a465

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Link Presentations

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

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	$-q^{8}+4q^{7}-9q^{6}+13q^{5}-16q^{4}+19q^{3}+q^{-3}-17q^{2}-2q^{-2}+15q+6q^{-1}-9$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{-6}-z^{2}a^{-6}-a^{-6}+z^{6}a^{-4}+2z^{4}a^{-4}+2z^{2}a^{-4}+a^{-4}+z^{6}a^{-2}+2z^{4}a^{-2}+a^{2}z^{2}+3z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}+2a^{2}+3a^{-2}-2z^{4}-5z^{2}-2z^{-2}-5$ (db)
Kauffman polynomial	$z^{10}a^{-2}+z^{10}a^{-4}+2z^{9}a^{-1}+7z^{9}a^{-3}+5z^{9}a^{-5}+5z^{8}a^{-2}+12z^{8}a^{-4}+9z^{8}a^{-6}+2z^{8}+2az^{7}+z^{7}a^{-1}-11z^{7}a^{-3}-2z^{7}a^{-5}+8z^{7}a^{-7}+a^{2}z^{6}-9z^{6}a^{-2}-31z^{6}a^{-4}-18z^{6}a^{-6}+4z^{6}a^{-8}+z^{6}-5az^{5}-5z^{5}a^{-1}+8z^{5}a^{-3}-7z^{5}a^{-5}-14z^{5}a^{-7}+z^{5}a^{-9}-4a^{2}z^{4}+31z^{4}a^{-4}+15z^{4}a^{-6}-5z^{4}a^{-8}-15z^{4}+2az^{3}-4z^{3}a^{-1}-5z^{3}a^{-3}+8z^{3}a^{-5}+6z^{3}a^{-7}-z^{3}a^{-9}+6a^{2}z^{2}+7z^{2}a^{-2}-13z^{2}a^{-4}-7z^{2}a^{-6}+19z^{2}+3az+7za^{-1}+3za^{-3}-3za^{-5}-2za^{-7}-4a^{2}-6a^{-2}+2a^{-4}+2a^{-6}-9-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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

3

13

6

1

-5

11

7

3

4

9

6

-3

7

10

7

3

5

7

9

2

3

8

10

-2

1

5

11

6

-1

1

4

-3

1

5

4

-5

1

-1

-7

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=1$	$i=3$
$r=-4$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=0$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{8}$
$r=1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=3$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=4$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=6$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=7$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$