L10a174

L10a173

L10n1

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

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 X18,11,19,12 X10,3,11,4 X4,9,1,10 X14,7,15,8 X8,13,5,14 X20,15,17,16 X16,19,13,20 X12,17,9,18
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
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 1 -5                   4 1 -3 -7                 6     6 -9               4 4     0 -11             11 6       5 -13           10 14         4 -15         5 1           4 -17       1 10             9 -19     5 5               0 -21   1 6                 5 -23                       0 -25 1                     1

Integral Khovanov Homology (db, data source)

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

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

L10a173 L10n1