L11a41

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)^{3}\left(t(2)^{2}-3t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {15}{q^{9/2}}}-q^{7/2}+{\frac {21}{q^{7/2}}}+5q^{5/2}-{\frac {26}{q^{5/2}}}-12q^{3/2}+{\frac {26}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {3}{q^{13/2}}}+{\frac {8}{q^{11/2}}}+18{\sqrt {q}}-{\frac {24}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}(-z)-a^{7}z^{-1}+3a^{5}z^{3}+5a^{5}z+3a^{5}z^{-1}-3a^{3}z^{5}-7a^{3}z^{3}-7a^{3}z-3a^{3}z^{-1}+az^{7}+3az^{5}-z^{5}a^{-1}+5az^{3}-z^{3}a^{-1}+4az+2az^{-1}-za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-3a^{8}z^{4}+3a^{8}z^{2}-a^{8}+3a^{7}z^{7}-7a^{7}z^{5}+6a^{7}z^{3}-3a^{7}z+a^{7}z^{-1}+5a^{6}z^{8}-8a^{6}z^{6}+2a^{6}z^{4}+3a^{6}z^{2}-2a^{6}+5a^{5}z^{9}-a^{5}z^{7}-15a^{5}z^{5}+21a^{5}z^{3}-12a^{5}z+3a^{5}z^{-1}+2a^{4}z^{10}+15a^{4}z^{8}-40a^{4}z^{6}+32a^{4}z^{4}-8a^{4}z^{2}+14a^{3}z^{9}-12a^{3}z^{7}-23a^{3}z^{5}+z^{5}a^{-3}+33a^{3}z^{3}-16a^{3}z+3a^{3}z^{-1}+2a^{2}z^{10}+25a^{2}z^{8}-58a^{2}z^{6}+5z^{6}a^{-2}+39a^{2}z^{4}-3z^{4}a^{-2}-11a^{2}z^{2}+2a^{2}+9az^{9}+4az^{7}+12z^{7}a^{-1}-32az^{5}-16z^{5}a^{-1}+24az^{3}+6z^{3}a^{-1}-9az-2za^{-1}+2az^{-1}+a^{-1}z^{-1}+15z^{8}-22z^{6}+9z^{4}-3z^{2}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

4

-4

4

8

1

7

2

10

4

-6

0

14

8

6

-2

14

12

-2

-4

12

0

-6

9

14

5

-8

6

12

-6

-10

2

9

7

-12

1

6

-5

-14

2

-16

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-4$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-3$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=-2$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=-1$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{14}$	${\mathbb {Z} }^{14}$
$r=0$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{14}$
$r=1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$