L11a537

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

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {-t(1)t(4)^{2}t(3)^{2}+t(1)t(2)t(4)^{2}t(3)^{2}-2t(2)t(4)^{2}t(3)^{2}+t(4)^{2}t(3)^{2}-t(1)t(3)^{2}-t(2)t(3)^{2}+2t(1)t(4)t(3)^{2}-t(1)t(2)t(4)t(3)^{2}+3t(2)t(4)t(3)^{2}-2t(4)t(3)^{2}+t(3)^{2}+2t(1)t(4)^{2}t(3)-2t(1)t(2)t(4)^{2}t(3)+3t(2)t(4)^{2}t(3)-t(4)^{2}t(3)+3t(1)t(3)-t(1)t(2)t(3)+2t(2)t(3)-4t(1)t(4)t(3)+2t(1)t(2)t(4)t(3)-4t(2)t(4)t(3)+2t(4)t(3)-2t(3)-t(1)t(4)^{2}+t(1)t(2)t(4)^{2}-t(2)t(4)^{2}-2t(1)+t(1)t(2)-t(2)+3t(1)t(4)-2t(1)t(2)t(4)+2t(2)t(4)-t(4)+1}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)t(4)}}$ (db)
Jones polynomial	$q^{11/2}-4q^{9/2}+9q^{7/2}-14q^{5/2}+17q^{3/2}-21{\sqrt {q}}+{\frac {17}{\sqrt {q}}}-{\frac {17}{q^{3/2}}}+{\frac {10}{q^{5/2}}}-{\frac {7}{q^{7/2}}}+{\frac {2}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+3az^{5}-4z^{5}a^{-1}+z^{5}a^{-3}-3a^{3}z^{3}+10az^{3}-8z^{3}a^{-1}+2z^{3}a^{-3}+a^{5}z-8a^{3}z+14az-9za^{-1}+2za^{-3}+2a^{5}z^{-1}-8a^{3}z^{-1}+11az^{-1}-6a^{-1}z^{-1}+a^{-3}z^{-1}+a^{5}z^{-3}-3a^{3}z^{-3}+3az^{-3}-a^{-1}z^{-3}$ (db)
Kauffman polynomial	$z^{4}a^{-6}+a^{5}z^{7}-5a^{5}z^{5}+4z^{5}a^{-5}+10a^{5}z^{3}-z^{3}a^{-5}-a^{5}z^{-3}-10a^{5}z+5a^{5}z^{-1}+2a^{4}z^{8}-6a^{4}z^{6}+9z^{6}a^{-4}+3a^{4}z^{4}-8z^{4}a^{-4}+7a^{4}z^{2}+3z^{2}a^{-4}+3a^{4}z^{-2}-9a^{4}-a^{-4}+2a^{3}z^{9}+a^{3}z^{7}+13z^{7}a^{-3}-24a^{3}z^{5}-19z^{5}a^{-3}+45a^{3}z^{3}+13z^{3}a^{-3}-3a^{3}z^{-3}-35a^{3}z-6za^{-3}+14a^{3}z^{-1}+2a^{-3}z^{-1}+a^{2}z^{10}+7a^{2}z^{8}+11z^{8}a^{-2}-23a^{2}z^{6}-10z^{6}a^{-2}+6a^{2}z^{4}-10z^{4}a^{-2}+24a^{2}z^{2}+14z^{2}a^{-2}+6a^{2}z^{-2}-21a^{2}-6a^{-2}+7az^{9}+5z^{9}a^{-1}-az^{7}+12z^{7}a^{-1}-48az^{5}-52z^{5}a^{-1}+74az^{3}+53z^{3}a^{-1}-3az^{-3}-a^{-1}z^{-3}-49az-30za^{-1}+18az^{-1}+11a^{-1}z^{-1}+z^{10}+16z^{8}-36z^{6}+2z^{4}+28z^{2}+3z^{-2}-18$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

12

1

-1

10

3

8

6

1

-5

6

8

3

5

4

10

7

-3

2

11

7

4

0

9

13

4

-2

8

0

-4

5

12

7

-6

2

5

-3

-8

1

6

5

-10

1

-1

-12

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{2}$
$r=-3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-2$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{8}$
$r=-1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=0$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{11}$
$r=1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=2$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{8}$
$r=3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$