L11a123

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

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(2)^{5}-2t(2)^{5}-4t(1)t(2)^{4}+4t(2)^{4}+4t(1)t(2)^{3}-4t(2)^{3}-4t(1)t(2)^{2}+4t(2)^{2}+4t(1)t(2)-4t(2)-2t(1)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{\sqrt {q}}}+{\frac {3}{q^{3/2}}}-{\frac {5}{q^{5/2}}}+{\frac {7}{q^{7/2}}}-{\frac {11}{q^{9/2}}}+{\frac {11}{q^{11/2}}}-{\frac {12}{q^{13/2}}}+{\frac {10}{q^{15/2}}}-{\frac {8}{q^{17/2}}}+{\frac {5}{q^{19/2}}}-{\frac {2}{q^{21/2}}}+{\frac {1}{q^{23/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{11}(-z)-2a^{11}z^{-1}+3a^{9}z^{3}+9a^{9}z+5a^{9}z^{-1}-3a^{7}z^{5}-11a^{7}z^{3}-10a^{7}z-3a^{7}z^{-1}+a^{5}z^{7}+4a^{5}z^{5}+4a^{5}z^{3}+a^{5}z-a^{3}z^{5}-3a^{3}z^{3}-a^{3}z$ (db)
Kauffman polynomial	$-z^{4}a^{14}+2z^{2}a^{14}-a^{14}-2z^{5}a^{13}+2z^{3}a^{13}-3z^{6}a^{12}+2z^{4}a^{12}-4z^{7}a^{11}+6z^{5}a^{11}-8z^{3}a^{11}+5za^{11}-2a^{11}z^{-1}-3z^{8}a^{10}+9z^{4}a^{10}-13z^{2}a^{10}+5a^{10}-2z^{9}a^{9}-2z^{7}a^{9}+14z^{5}a^{9}-20z^{3}a^{9}+15za^{9}-5a^{9}z^{-1}-z^{10}a^{8}-2z^{8}a^{8}+6z^{6}a^{8}+4z^{4}a^{8}-9z^{2}a^{8}+5a^{8}-5z^{9}a^{7}+14z^{7}a^{7}-8z^{5}a^{7}-4z^{3}a^{7}+9za^{7}-3a^{7}z^{-1}-z^{10}a^{6}-2z^{8}a^{6}+16z^{6}a^{6}-17z^{4}a^{6}+5z^{2}a^{6}-3z^{9}a^{5}+11z^{7}a^{5}-10z^{5}a^{5}+2z^{3}a^{5}-3z^{8}a^{4}+13z^{6}a^{4}-15z^{4}a^{4}+3z^{2}a^{4}-z^{7}a^{3}+4z^{5}a^{3}-4z^{3}a^{3}+za^{3}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

0

1

-2

2

-2

-4

3

1

2

-6

5

3

-2

-8

6

2

4

-10

5

0

-12

7

6

1

-14

4

6

2

-16

4

6

-2

-18

1

4

3

-20

1

4

-3

-22

1

-24

1

-1

Integral Khovanov Homology

(db, data source)

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