L10a174

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a174 at Knotilus!
	[edit L10a174 Quick Notes] L10a174 is a closed five-link chain.

[edit L10a174 Further Notes and Views]

Five linked circles.		Decorative pentagonal representation.		(alternate version)		Highly ornamental version.
Quasi-yin-yang.

Link Presentations

[edit Notes on L10a174's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uvwx+uvwy-uvw+uvxy-uvx-2uvy+uv+uwxy-2uwx-uwy+uw-2uxy+2ux+2uy-u+vwxy-2vwx-2vwy+2vw-vxy+vx+2vy-v-wxy+2wx+wy-w+xy-x-y}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}{\sqrt {x}}{\sqrt {y}}}}$ (db)
Jones polynomial	$q^{-12}-q^{-11}+6q^{-10}-6q^{-9}+15q^{-8}-11q^{-7}+15q^{-6}-10q^{-5}+10q^{-4}-4q^{-3}+q^{-2}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{14}z^{-4}-4a^{12}z^{-4}-5a^{12}z^{-2}+6a^{10}z^{-4}+15a^{10}z^{-2}+10a^{10}-4a^{8}z^{-4}-10a^{8}z^{2}-15a^{8}z^{-2}-20a^{8}+4a^{6}z^{4}+a^{6}z^{-4}+10a^{6}z^{2}+5a^{6}z^{-2}+10a^{6}+a^{4}z^{4}$ (db)
Kauffman polynomial	$a^{14}z^{6}-5a^{14}z^{4}-a^{14}z^{-4}+10a^{14}z^{2}+5a^{14}z^{-2}-10a^{14}+a^{13}z^{7}-10a^{13}z^{3}+4a^{13}z^{-3}+20a^{13}z-15a^{13}z^{-1}+a^{12}z^{8}+4a^{12}z^{6}-20a^{12}z^{4}-4a^{12}z^{-4}+30a^{12}z^{2}+14a^{12}z^{-2}-25a^{12}+a^{11}z^{9}+2a^{11}z^{7}+2a^{11}z^{5}-30a^{11}z^{3}+12a^{11}z^{-3}+55a^{11}z-41a^{11}z^{-1}+6a^{10}z^{8}-2a^{10}z^{6}-25a^{10}z^{4}-6a^{10}z^{-4}+40a^{10}z^{2}+18a^{10}z^{-2}-31a^{10}+a^{9}z^{9}+11a^{9}z^{7}-12a^{9}z^{5}-30a^{9}z^{3}+12a^{9}z^{-3}+55a^{9}z-41a^{9}z^{-1}+5a^{8}z^{8}+5a^{8}z^{6}-25a^{8}z^{4}-4a^{8}z^{-4}+30a^{8}z^{2}+14a^{8}z^{-2}-25a^{8}+10a^{7}z^{7}-10a^{7}z^{5}-10a^{7}z^{3}+4a^{7}z^{-3}+20a^{7}z-15a^{7}z^{-1}+10a^{6}z^{6}-14a^{6}z^{4}-a^{6}z^{-4}+10a^{6}z^{2}+5a^{6}z^{-2}-10a^{6}+4a^{5}z^{5}+a^{4}z^{4}$ (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		\

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-3

1

-5

4

1

-3

-7

6

-9

4

0

-11

11

6

5

-13

10

14

4

-15

5

1

4

-17

1

10

9

-19

5

0

-21

1

6

5

-23

0

-25

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-5$	$i=-3$
$r=-10$	${\mathbb {Z} }$
$r=-9$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-8$	${\mathbb {Z} }^{6}$	${\mathbb {Z} }^{5}$
$r=-7$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-4$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{11}$
$r=-3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-1$	${\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$