L11a157

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_10,4,11,3 X_22,10,7,9 X₂₇₃₈ X_18,13,19,14 X_6,12,1,11 X_20,18,21,17 X_16,22,17,21 X_4,16,5,15 X_14,6,15,5 X_12,19,13,20
Gauss code	{1, -4, 2, -9, 10, -6}, {4, -1, 3, -2, 6, -11, 5, -10, 9, -8, 7, -5, 11, -7, 8, -3}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(2)^{4}-t(2)^{4}+2t(1)^{2}t(2)^{3}-8t(1)t(2)^{3}+5t(2)^{3}-6t(1)^{2}t(2)^{2}+15t(1)t(2)^{2}-6t(2)^{2}+5t(1)^{2}t(2)-8t(1)t(2)+2t(2)-t(1)^{2}+t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{15/2}-4q^{13/2}+8q^{11/2}-13q^{9/2}+17q^{7/2}-20q^{5/2}+19q^{3/2}-17{\sqrt {q}}+{\frac {12}{\sqrt {q}}}-{\frac {7}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {1}{q^{7/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$za^{-7}-3z^{3}a^{-5}-3za^{-5}-a^{-5}z^{-1}+2z^{5}a^{-3}+4z^{3}a^{-3}+a^{3}z+6za^{-3}+2a^{-3}z^{-1}+z^{5}a^{-1}-2az^{3}-2z^{3}a^{-1}-4za^{-1}+az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-2}-z^{10}a^{-4}-4z^{9}a^{-1}-8z^{9}a^{-3}-4z^{9}a^{-5}-14z^{8}a^{-2}-14z^{8}a^{-4}-6z^{8}a^{-6}-6z^{8}-5az^{7}-3z^{7}a^{-1}+4z^{7}a^{-3}-2z^{7}a^{-5}-4z^{7}a^{-7}-3a^{2}z^{6}+36z^{6}a^{-2}+39z^{6}a^{-4}+13z^{6}a^{-6}-z^{6}a^{-8}+8z^{6}-a^{3}z^{5}+7az^{5}+18z^{5}a^{-1}+25z^{5}a^{-3}+25z^{5}a^{-5}+10z^{5}a^{-7}+5a^{2}z^{4}-34z^{4}a^{-2}-31z^{4}a^{-4}-6z^{4}a^{-6}+2z^{4}a^{-8}-6z^{4}+2a^{3}z^{3}-4az^{3}-24z^{3}a^{-1}-35z^{3}a^{-3}-24z^{3}a^{-5}-7z^{3}a^{-7}-2a^{2}z^{2}+11z^{2}a^{-2}+8z^{2}a^{-4}+z^{2}a^{-6}-z^{2}a^{-8}+3z^{2}-a^{3}z+3az+13za^{-1}+15za^{-3}+8za^{-5}+2za^{-7}-a^{-2}-az^{-1}-2a^{-1}z^{-1}-2a^{-3}z^{-1}-a^{-5}z^{-1}$ (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

6

7

χ

16

1

-1

14

3

12

5

1

-4

10

8

3

5

8

9

5

-4

6

11

8

3

4

9

10

1

2

8

10

-2

0

5

10

5

-2

2

7

-5

-4

1

5

4

-6

2

-2

-8

1

Integral Khovanov Homology

(db, data source)

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