L11a298

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_12,3,13,4 X_18,8,19,7 X_16,6,17,5 X_22,13,9,14 X_20,15,21,16 X_6,18,7,17 X_14,21,15,22 X_4,20,5,19 X_2,9,3,10 X_8,11,1,12
Gauss code	{1, -10, 2, -9, 4, -7, 3, -11}, {10, -1, 11, -2, 5, -8, 6, -4, 7, -3, 9, -6, 8, -5}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(2t(2)t(1)^{2}+2t(2)^{2}t(1)-t(2)t(1)+2t(1)+2t(2)\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-q^{9/2}+3q^{7/2}-6q^{5/2}+8q^{3/2}-10{\sqrt {q}}+{\frac {11}{\sqrt {q}}}-{\frac {11}{q^{3/2}}}+{\frac {8}{q^{5/2}}}-{\frac {7}{q^{7/2}}}+{\frac {4}{q^{9/2}}}-{\frac {2}{q^{11/2}}}+{\frac {1}{q^{13/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{5}-2za^{5}+z^{5}a^{3}+2z^{3}a^{3}+2z^{5}a+6z^{3}a+5za+az^{-1}+z^{5}a^{-1}+z^{3}a^{-1}-2za^{-1}-a^{-1}z^{-1}-z^{3}a^{-3}-za^{-3}$ (db)
Kauffman polynomial	$-a^{4}z^{10}-a^{2}z^{10}-2a^{5}z^{9}-5a^{3}z^{9}-3az^{9}-a^{6}z^{8}+a^{4}z^{8}-4a^{2}z^{8}-6z^{8}+11a^{5}z^{7}+19a^{3}z^{7}-8z^{7}a^{-1}+6a^{6}z^{6}+11a^{4}z^{6}+23a^{2}z^{6}-8z^{6}a^{-2}+10z^{6}-20a^{5}z^{5}-17a^{3}z^{5}+23az^{5}+14z^{5}a^{-1}-6z^{5}a^{-3}-12a^{6}z^{4}-22a^{4}z^{4}-22a^{2}z^{4}+11z^{4}a^{-2}-3z^{4}a^{-4}+2z^{4}+14a^{5}z^{3}-26az^{3}-6z^{3}a^{-1}+5z^{3}a^{-3}-z^{3}a^{-5}+8a^{6}z^{2}+9a^{4}z^{2}+4a^{2}z^{2}-4z^{2}a^{-2}-z^{2}-4a^{5}z+10az+4za^{-1}-2za^{-3}+1-az^{-1}-a^{-1}z^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

10

1

8

2

-2

6

4

1

3

4

2

-2

2

6

4

2

0

7

6

-1

-2

4

0

-4

4

7

3

-6

3

4

-1

-8

1

4

3

-10

1

3

-2

-12

1

-14

1

-1

Integral Khovanov Homology

(db, data source)

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