L11n372

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(3)-1)\left(-t(1)t(3)^{3}+t(1)t(2)t(3)^{3}+t(1)t(2)^{2}t(3)^{2}+t(1)t(3)^{2}-2t(1)t(2)t(3)^{2}-t(3)^{2}-t(1)t(2)^{2}t(3)+t(2)^{2}t(3)-2t(2)t(3)+t(3)-t(2)^{2}+t(2)\right)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$-q^{4}+3q^{3}-5q^{2}+8q-8+10q^{-1}-8q^{-2}+7q^{-3}-4q^{-4}+2q^{-5}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{6}-a^{4}z^{4}-3a^{4}z^{2}+a^{4}z^{-2}-a^{4}+a^{2}z^{6}+3a^{2}z^{4}-z^{4}a^{-2}+a^{2}z^{2}-2a^{2}z^{-2}-2z^{2}a^{-2}-2a^{2}+z^{6}+3z^{4}+2z^{2}+z^{-2}+2$ (db)
Kauffman polynomial	$3a^{6}z^{2}-a^{6}+a^{5}z^{5}+3a^{5}z^{3}-a^{5}z+3a^{4}z^{6}-a^{4}z^{4}-3a^{4}z^{2}-a^{4}z^{-2}+4a^{4}+6a^{3}z^{7}+z^{7}a^{-3}-16a^{3}z^{5}-4z^{5}a^{-3}+17a^{3}z^{3}+5z^{3}a^{-3}-10a^{3}z-2za^{-3}+2a^{3}z^{-1}+6a^{2}z^{8}+3z^{8}a^{-2}-20a^{2}z^{6}-13z^{6}a^{-2}+27a^{2}z^{4}+17z^{4}a^{-2}-26a^{2}z^{2}-8z^{2}a^{-2}-2a^{2}z^{-2}+12a^{2}+2a^{-2}+2az^{9}+2z^{9}a^{-1}+2az^{7}-3z^{7}a^{-1}-25az^{5}-12z^{5}a^{-1}+30az^{3}+21z^{3}a^{-1}-14az-7za^{-1}+2az^{-1}+9z^{8}-36z^{6}+45z^{4}-28z^{2}-z^{-2}+10$ (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

χ

9

1

-1

7

2

5

3

1

-2

3

5

2

3

1

4

0

-1

6

4

2

-3

3

5

2

-5

4

5

-1

-7

1

4

3

-9

1

3

-2

-11

2

Integral Khovanov Homology

(db, data source)

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