L11a420

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

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)t(2)^{2}t(3)^{4}-t(2)^{2}t(3)^{4}-t(1)t(2)t(3)^{4}-2t(1)t(2)^{2}t(3)^{3}+3t(2)^{2}t(3)^{3}-t(1)t(3)^{3}+3t(1)t(2)t(3)^{3}-2t(2)t(3)^{3}+2t(1)t(2)^{2}t(3)^{2}-3t(2)^{2}t(3)^{2}+3t(1)t(3)^{2}-4t(1)t(2)t(3)^{2}+4t(2)t(3)^{2}-2t(3)^{2}+t(2)^{2}t(3)-3t(1)t(3)+2t(1)t(2)t(3)-3t(2)t(3)+2t(3)+t(1)+t(2)-1}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$q^{7}-3q^{6}+7q^{5}-10q^{4}-q^{-4}+14q^{3}+2q^{-3}-14q^{2}-5q^{-2}+15q+9q^{-1}-11$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{8}a^{-2}-6z^{6}a^{-2}+z^{6}a^{-4}+2z^{6}-a^{2}z^{4}-15z^{4}a^{-2}+4z^{4}a^{-4}+10z^{4}-4a^{2}z^{2}-21z^{2}a^{-2}+6z^{2}a^{-4}+18z^{2}-4a^{2}-16a^{-2}+5a^{-4}+15-a^{2}z^{-2}-5a^{-2}z^{-2}+2a^{-4}z^{-2}+4z^{-2}$ (db)
Kauffman polynomial	$z^{10}a^{-2}+z^{10}+2az^{9}+6z^{9}a^{-1}+4z^{9}a^{-3}+2a^{2}z^{8}+9z^{8}a^{-2}+7z^{8}a^{-4}+4z^{8}+a^{3}z^{7}-3az^{7}-12z^{7}a^{-1}-z^{7}a^{-3}+7z^{7}a^{-5}-8a^{2}z^{6}-35z^{6}a^{-2}-12z^{6}a^{-4}+6z^{6}a^{-6}-25z^{6}-5a^{3}z^{5}-12az^{5}-11z^{5}a^{-1}-15z^{5}a^{-3}-8z^{5}a^{-5}+3z^{5}a^{-7}+10a^{2}z^{4}+47z^{4}a^{-2}+11z^{4}a^{-4}-8z^{4}a^{-6}+z^{4}a^{-8}+37z^{4}+8a^{3}z^{3}+28az^{3}+39z^{3}a^{-1}+23z^{3}a^{-3}+2z^{3}a^{-5}-2z^{3}a^{-7}-7a^{2}z^{2}-39z^{2}a^{-2}-9z^{2}a^{-4}+6z^{2}a^{-6}-z^{2}a^{-8}-30z^{2}-5a^{3}z-21az-33za^{-1}-16za^{-3}+za^{-5}+4a^{2}+20a^{-2}+6a^{-4}-2a^{-6}+17+a^{3}z^{-1}+5az^{-1}+9a^{-1}z^{-1}+5a^{-3}z^{-1}-a^{2}z^{-2}-5a^{-2}z^{-2}-2a^{-4}z^{-2}-4z^{-2}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

15

1

13

3

1

-2

11

4

9

6

3

-3

7

8

4

5

7

0

3

8

7

1

5

9

4

-1

4

6

-2

-3

1

5

4

-5

1

4

-3

-7

1

-9

1

-1

Integral Khovanov Homology

(db, data source)

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