L11n297

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_15,20,16,21 X_14,8,15,7 X_21,10,22,5 X_18,11,19,12 X_9,17,10,16 X_22,17,11,18 X_8,19,9,20 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -9, -7, 5}, {6, -2, 11, -4, -3, 7, 8, -6, 9, 3, -5, -8}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)t(3)^{2}t(2)^{2}-t(3)^{2}t(2)^{2}-t(1)t(3)t(2)^{2}+2t(3)t(2)^{2}-t(2)^{2}-t(1)t(3)^{2}t(2)+2t(1)t(3)t(2)-2t(3)t(2)+t(2)+t(1)t(3)^{2}+t(1)-2t(1)t(3)+t(3)-1}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$q^{2}-2q+5-4q^{-1}+7q^{-2}-5q^{-3}+5q^{-4}-4q^{-5}+2q^{-6}-q^{-7}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{2}a^{6}-a^{6}z^{-2}-2a^{6}+2z^{4}a^{4}+7z^{2}a^{4}+4a^{4}z^{-2}+8a^{4}-z^{6}a^{2}-5z^{4}a^{2}-10z^{2}a^{2}-5a^{2}z^{-2}-10a^{2}+z^{4}+3z^{2}+2z^{-2}+4$ (db)
Kauffman polynomial	$a^{5}z^{9}+a^{3}z^{9}+2a^{6}z^{8}+5a^{4}z^{8}+3a^{2}z^{8}+a^{7}z^{7}+a^{3}z^{7}+2az^{7}-9a^{6}z^{6}-22a^{4}z^{6}-13a^{2}z^{6}-5a^{7}z^{5}-16a^{5}z^{5}-17a^{3}z^{5}-6az^{5}+11a^{6}z^{4}+30a^{4}z^{4}+24a^{2}z^{4}+5z^{4}+8a^{7}z^{3}+29a^{5}z^{3}+29a^{3}z^{3}+10az^{3}+2z^{3}a^{-1}-5a^{6}z^{2}-22a^{4}z^{2}-27a^{2}z^{2}+z^{2}a^{-2}-9z^{2}-5a^{7}z-18a^{5}z-24a^{3}z-11az+3a^{6}+12a^{4}+15a^{2}+7+a^{7}z^{-1}+5a^{5}z^{-1}+9a^{3}z^{-1}+5az^{-1}-a^{6}z^{-2}-4a^{4}z^{-2}-5a^{2}z^{-2}-2z^{-2}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

5

1

3

2

1

-1

1

3

-1

3

4

1

-3

4

1

3

-5

2

4

2

-7

3

0

-9

1

2

1

-11

1

3

-2

-13

1

-15

1

-1

Integral Khovanov Homology

(db, data source)

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