L11n371

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(3)-1)(t(1)t(3)+1)(t(3)t(2)-t(2)+1)(t(2)t(3)-t(3)+1)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$q^{3}-3q^{2}+7q-9+12q^{-1}-12q^{-2}+12q^{-3}-8q^{-4}+6q^{-5}-2q^{-6}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{6}z^{2}-2a^{6}+a^{4}z^{6}+5a^{4}z^{4}+10a^{4}z^{2}+a^{4}z^{-2}+8a^{4}-a^{2}z^{8}-6a^{2}z^{6}-14a^{2}z^{4}-17a^{2}z^{2}-2a^{2}z^{-2}-11a^{2}+z^{6}+4z^{4}+6z^{2}+z^{-2}+5$ (db)
Kauffman polynomial	$3a^{7}z^{3}-2a^{7}z+a^{6}z^{6}+6a^{6}z^{4}-8a^{6}z^{2}+2a^{6}+5a^{5}z^{7}-11a^{5}z^{5}+17a^{5}z^{3}-7a^{5}z+6a^{4}z^{8}-18a^{4}z^{6}+33a^{4}z^{4}-28a^{4}z^{2}-a^{4}z^{-2}+10a^{4}+2a^{3}z^{9}+5a^{3}z^{7}-24a^{3}z^{5}+30a^{3}z^{3}-14a^{3}z+2a^{3}z^{-1}+10a^{2}z^{8}-29a^{2}z^{6}+z^{6}a^{-2}+33a^{2}z^{4}-3z^{4}a^{-2}-26a^{2}z^{2}+3z^{2}a^{-2}-2a^{2}z^{-2}+12a^{2}-a^{-2}+2az^{9}+3az^{7}+3z^{7}a^{-1}-21az^{5}-8z^{5}a^{-1}+21az^{3}+5z^{3}a^{-1}-10az-za^{-1}+2az^{-1}+4z^{8}-9z^{6}+3z^{4}-3z^{2}-z^{-2}+4$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

χ

7

1

5

2

-2

3

5

1

4

1

5

3

-2

-1

7

4

3

-3

6

0

-5

6

0

-7

3

7

4

-9

3

5

-2

-11

4

-13

2

-2

Integral Khovanov Homology

(db, data source)

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