L11a225

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a225 at Knotilus!
	[edit L11a225 Quick Notes]

[edit L11a225 Further Notes and Views]

Link Presentations

[edit Notes on L11a225's Link Presentations]

Planar diagram presentation	X₈₁₉₂ X_16,6,17,5 X_18,10,19,9 X_10,20,11,19 X_2,11,3,12 X_12,3,13,4 X₄₇₅₈ X_20,16,21,15 X_22,14,7,13 X_14,22,15,21 X_6,18,1,17
Gauss code	{1, -5, 6, -7, 2, -11}, {7, -1, 3, -4, 5, -6, 9, -10, 8, -2, 11, -3, 4, -8, 10, -9}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2u^{2}v^{4}-4u^{2}v^{3}+4u^{2}v^{2}-2u^{2}v-2uv^{4}+8uv^{3}-11uv^{2}+8uv-2u-2v^{3}+4v^{2}-4v+2}{uv^{2}}}$ (db)
Jones polynomial	$q^{15/2}-3q^{13/2}+7q^{11/2}-11q^{9/2}+15q^{7/2}-18q^{5/2}+17q^{3/2}-16{\sqrt {q}}+{\frac {11}{\sqrt {q}}}-{\frac {7}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {1}{q^{7/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-5}+3z^{3}a^{-5}+3za^{-5}+a^{-5}z^{-1}-z^{7}a^{-3}-4z^{5}a^{-3}-6z^{3}a^{-3}-4za^{-3}-2a^{-3}z^{-1}-z^{7}a^{-1}+az^{5}-4z^{5}a^{-1}+3az^{3}-6z^{3}a^{-1}+3az-3za^{-1}+az^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-8}-3z^{4}a^{-8}+2z^{2}a^{-8}+3z^{7}a^{-7}-8z^{5}a^{-7}+4z^{3}a^{-7}+5z^{8}a^{-6}-14z^{6}a^{-6}+13z^{4}a^{-6}-9z^{2}a^{-6}+2a^{-6}+5z^{9}a^{-5}-13z^{7}a^{-5}+14z^{5}a^{-5}-12z^{3}a^{-5}+4za^{-5}-a^{-5}z^{-1}+2z^{10}a^{-4}+4z^{8}a^{-4}-24z^{6}a^{-4}+35z^{4}a^{-4}-22z^{2}a^{-4}+5a^{-4}+10z^{9}a^{-3}-28z^{7}a^{-3}+a^{3}z^{5}+32z^{5}a^{-3}-2a^{3}z^{3}-14z^{3}a^{-3}+a^{3}z+5za^{-3}-2a^{-3}z^{-1}+2z^{10}a^{-2}+5z^{8}a^{-2}+3a^{2}z^{6}-22z^{6}a^{-2}-5a^{2}z^{4}+30z^{4}a^{-2}+a^{2}z^{2}-13z^{2}a^{-2}+3a^{-2}+5z^{9}a^{-1}+5az^{7}-7z^{7}a^{-1}-8az^{5}+z^{5}a^{-1}+4az^{3}+8z^{3}a^{-1}-4az-4za^{-1}+az^{-1}+6z^{8}-10z^{6}+6z^{4}-z^{2}-1$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

16

1

-1

14

2

12

5

1

-4

10

6

2

4

8

9

5

-4

6

9

6

3

4

8

9

1

2

8

9

-1

0

4

9

5

-2

3

7

-4

1

5

4

-6

2

-2

-8

1

Integral Khovanov Homology

(db, data source)

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