L11a415

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-t(1)t(3)^{4}+t(1)t(2)t(3)^{4}-2t(2)t(3)^{4}+t(3)^{4}+t(1)t(2)^{2}t(3)^{3}-2t(2)^{2}t(3)^{3}+2t(1)t(3)^{3}-3t(1)t(2)t(3)^{3}+4t(2)t(3)^{3}-t(3)^{3}-t(1)t(2)^{2}t(3)^{2}+2t(2)^{2}t(3)^{2}-2t(1)t(3)^{2}+4t(1)t(2)t(3)^{2}-4t(2)t(3)^{2}+t(3)^{2}+t(1)t(2)^{2}t(3)-2t(2)^{2}t(3)+2t(1)t(3)-4t(1)t(2)t(3)+3t(2)t(3)-t(3)-t(1)t(2)^{2}+t(2)^{2}+2t(1)t(2)-t(2)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$-q^{8}+3q^{7}-7q^{6}+11q^{5}-14q^{4}+17q^{3}+q^{-3}-15q^{2}-2q^{-2}+14q+6q^{-1}-9$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{-6}-2z^{2}a^{-6}-a^{-6}z^{-2}-2a^{-6}+z^{6}a^{-4}+3z^{4}a^{-4}+6z^{2}a^{-4}+4a^{-4}z^{-2}+8a^{-4}+z^{6}a^{-2}+z^{4}a^{-2}+a^{2}z^{2}-4z^{2}a^{-2}-5a^{-2}z^{-2}+2a^{2}-8a^{-2}-2z^{4}-4z^{2}+2z^{-2}$ (db)
Kauffman polynomial	$z^{10}a^{-2}+z^{10}a^{-4}+2z^{9}a^{-1}+6z^{9}a^{-3}+4z^{9}a^{-5}+5z^{8}a^{-2}+9z^{8}a^{-4}+7z^{8}a^{-6}+3z^{8}+2az^{7}+2z^{7}a^{-1}-8z^{7}a^{-3}-2z^{7}a^{-5}+6z^{7}a^{-7}+a^{2}z^{6}-17z^{6}a^{-2}-30z^{6}a^{-4}-18z^{6}a^{-6}+3z^{6}a^{-8}-7z^{6}-5az^{5}-16z^{5}a^{-1}-9z^{5}a^{-3}-13z^{5}a^{-5}-14z^{5}a^{-7}+z^{5}a^{-9}-4a^{2}z^{4}+27z^{4}a^{-2}+44z^{4}a^{-4}+22z^{4}a^{-6}-5z^{4}a^{-8}+6z^{4}+2az^{3}+21z^{3}a^{-1}+35z^{3}a^{-3}+30z^{3}a^{-5}+12z^{3}a^{-7}-2z^{3}a^{-9}+5a^{2}z^{2}-33z^{2}a^{-2}-33z^{2}a^{-4}-12z^{2}a^{-6}-7z^{2}+az-16za^{-1}-33za^{-3}-21za^{-5}-5za^{-7}-2a^{2}+20a^{-2}+17a^{-4}+4a^{-6}+6+5a^{-1}z^{-1}+9a^{-3}z^{-1}+5a^{-5}z^{-1}+a^{-7}z^{-1}-5a^{-2}z^{-2}-4a^{-4}z^{-2}-a^{-6}z^{-2}-2z^{-2}$ (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

2

13

5

1

-4

11

6

2

4

9

8

5

-3

7

9

6

3

5

8

10

2

3

6

7

-1

1

4

9

5

-1

2

5

-3

4

-5

1

2

-1

-7

1

Integral Khovanov Homology

(db, data source)

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