L11a153

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{2}v^{5}-2u^{2}v^{4}+3u^{2}v^{3}-3u^{2}v^{2}+2u^{2}v+uv^{6}-3uv^{5}+6uv^{4}-7uv^{3}+6uv^{2}-3uv+u+2v^{5}-3v^{4}+3v^{3}-2v^{2}+v}{uv^{3}}}$ (db)
Jones polynomial	$-3q^{9/2}+{\frac {2}{q^{9/2}}}+7q^{7/2}-{\frac {6}{q^{7/2}}}-11q^{5/2}+{\frac {9}{q^{5/2}}}+14q^{3/2}-{\frac {13}{q^{3/2}}}+q^{11/2}-{\frac {1}{q^{11/2}}}-16{\sqrt {q}}+{\frac {15}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-az^{7}-z^{7}a^{-1}+a^{3}z^{5}-5az^{5}-4z^{5}a^{-1}+z^{5}a^{-3}+4a^{3}z^{3}-11az^{3}-5z^{3}a^{-1}+3z^{3}a^{-3}+6a^{3}z-12az+3za^{-3}+3a^{3}z^{-1}-5az^{-1}+2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{2}z^{10}-z^{10}-3a^{3}z^{9}-7az^{9}-4z^{9}a^{-1}-2a^{4}z^{8}-5a^{2}z^{8}-7z^{8}a^{-2}-10z^{8}-a^{5}z^{7}+11a^{3}z^{7}+19az^{7}-z^{7}a^{-1}-8z^{7}a^{-3}+7a^{4}z^{6}+26a^{2}z^{6}+8z^{6}a^{-2}-6z^{6}a^{-4}+33z^{6}+5a^{5}z^{5}-16a^{3}z^{5}-22az^{5}+14z^{5}a^{-1}+12z^{5}a^{-3}-3z^{5}a^{-5}-6a^{4}z^{4}-33a^{2}z^{4}+7z^{4}a^{-4}-z^{4}a^{-6}-35z^{4}-8a^{5}z^{3}+17a^{3}z^{3}+29az^{3}-8z^{3}a^{-1}-10z^{3}a^{-3}+2z^{3}a^{-5}+19a^{2}z^{2}-3z^{2}a^{-2}-5z^{2}a^{-4}+z^{2}a^{-6}+22z^{2}+4a^{5}z-12a^{3}z-21az-2za^{-1}+3za^{-3}-5a^{2}+a^{-4}-5+3a^{3}z^{-1}+5az^{-1}+2a^{-1}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

12

1

-1

10

2

8

5

1

-4

6

2

4

8

5

-3

2

8

6

2

0

8

9

1

-2

5

7

-2

-4

4

8

4

-6

2

5

-3

-8

4

-10

1

2

-1

-12

1

Integral Khovanov Homology

(db, data source)

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