L11a28

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Link Presentations

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

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	$-10q^{9/2}+{\frac {1}{q^{9/2}}}+17q^{7/2}-{\frac {4}{q^{7/2}}}-23q^{5/2}+{\frac {9}{q^{5/2}}}+26q^{3/2}-{\frac {17}{q^{3/2}}}-q^{13/2}+4q^{11/2}-26{\sqrt {q}}+{\frac {22}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-5}-za^{-5}-a^{-5}z^{-1}+2z^{5}a^{-3}-a^{3}z^{3}+4z^{3}a^{-3}-a^{3}z+5za^{-3}+3a^{-3}z^{-1}-z^{7}a^{-1}+2az^{5}-3z^{5}a^{-1}+4az^{3}-6z^{3}a^{-1}+4az-7za^{-1}+2az^{-1}-4a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-z^{3}a^{-7}+4z^{6}a^{-6}-4z^{4}a^{-6}+z^{2}a^{-6}+9z^{7}a^{-5}-13z^{5}a^{-5}+10z^{3}a^{-5}-4za^{-5}+a^{-5}z^{-1}+12z^{8}a^{-4}+a^{4}z^{6}-18z^{6}a^{-4}-2a^{4}z^{4}+13z^{4}a^{-4}+a^{4}z^{2}-5z^{2}a^{-4}+a^{-4}+8z^{9}a^{-3}+4a^{3}z^{7}+4z^{7}a^{-3}-9a^{3}z^{5}-32z^{5}a^{-3}+8a^{3}z^{3}+34z^{3}a^{-3}-3a^{3}z-16za^{-3}+3a^{-3}z^{-1}+2z^{10}a^{-2}+7a^{2}z^{8}+24z^{8}a^{-2}-13a^{2}z^{6}-58z^{6}a^{-2}+7a^{2}z^{4}+45z^{4}a^{-2}-a^{2}z^{2}-17z^{2}a^{-2}+a^{2}+3a^{-2}+6az^{9}+14z^{9}a^{-1}+az^{7}-8z^{7}a^{-1}-27az^{5}-36z^{5}a^{-1}+30az^{3}+45z^{3}a^{-1}-13az-22za^{-1}+2az^{-1}+4a^{-1}z^{-1}+2z^{10}+19z^{8}-50z^{6}+37z^{4}-13z^{2}+2$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

3

-3

10

7

1

6

8

10

3

-7

6

13

7

6

4

13

10

-3

2

13

0

11

15

4

-2

6

11

-5

-4

3

11

8

-6

1

6

-5

-8

3

-10

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=0$	${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{13}$
$r=1$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=2$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=3$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$