L11a435

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(v-1)\left(-uvw^{3}+4uvw^{2}-4uvw+2uv+uw^{3}-2uw^{2}+2uw+2v^{2}w^{2}-2v^{2}w+v^{2}+2vw^{3}-4vw^{2}+4vw-v\right)}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$-q^{8}+3q^{7}-7q^{6}+13q^{5}-18q^{4}+21q^{3}-20q^{2}+19q-13+9q^{-1}-3q^{-2}+q^{-3}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-a^{-8}+3z^{2}a^{-6}-a^{-6}z^{-2}+2a^{-6}-3z^{4}a^{-4}-2z^{2}a^{-4}+4a^{-4}z^{-2}+4a^{-4}+z^{6}a^{-2}+a^{2}z^{2}-5z^{2}a^{-2}-5a^{-2}z^{-2}+a^{2}-9a^{-2}-2z^{4}-z^{2}+2z^{-2}+3$ (db)
Kauffman polynomial	$z^{10}a^{-2}+z^{10}a^{-4}+4z^{9}a^{-1}+8z^{9}a^{-3}+4z^{9}a^{-5}+16z^{8}a^{-2}+16z^{8}a^{-4}+6z^{8}a^{-6}+6z^{8}+3az^{7}+4z^{7}a^{-1}+3z^{7}a^{-3}+7z^{7}a^{-5}+5z^{7}a^{-7}+a^{2}z^{6}-46z^{6}a^{-2}-38z^{6}a^{-4}-5z^{6}a^{-6}+3z^{6}a^{-8}-15z^{6}-6az^{5}-32z^{5}a^{-1}-48z^{5}a^{-3}-29z^{5}a^{-5}-6z^{5}a^{-7}+z^{5}a^{-9}-3a^{2}z^{4}+52z^{4}a^{-2}+35z^{4}a^{-4}-z^{4}a^{-6}-5z^{4}a^{-8}+18z^{4}+3az^{3}+39z^{3}a^{-1}+68z^{3}a^{-3}+37z^{3}a^{-5}+3z^{3}a^{-7}-2z^{3}a^{-9}+3a^{2}z^{2}-44z^{2}a^{-2}-22z^{2}a^{-4}+4z^{2}a^{-6}+3z^{2}a^{-8}-18z^{2}-23za^{-1}-41za^{-3}-21za^{-5}-2za^{-7}+za^{-9}-a^{2}+23a^{-2}+14a^{-4}-a^{-8}+10+5a^{-1}z^{-1}+9a^{-3}z^{-1}+5a^{-5}z^{-1}+a^{-7}z^{-1}-5a^{-2}z^{-2}-4a^{-4}z^{-2}-a^{-6}z^{-2}-2z^{-2}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

17

1

-1

15

2

13

5

1

-4

11

8

2

6

9

11

6

-5

7

10

7

3

5

10

11

1

3

9

10

-1

1

6

12

6

-1

3

7

-4

-3

6

-5

1

3

-2

-7

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=1$	$i=3$
$r=-4$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=0$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{9}$
$r=1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=3$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=4$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{8}$
$r=5$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=6$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=7$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$