L10a101

L10a100

L10a102

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a101 at Knotilus!
	Contains two L4a1 configurations.

As a Celtic (or pseudo-Celtic) linear decorative knot

(multicolored)

Mongolian ornament

Rotated knotwork cross with four L10a101 sub-configurations

Link Presentations

[edit Notes on L10a101's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,3,13,4 X14,19,15,20 X18,7,19,8 X6,15,7,16 X16,5,17,6 X4,17,5,18 X20,13,9,14 X2,9,3,10 X8,11,1,12
Gauss code	{1, -9, 2, -7, 6, -5, 4, -10}, {9, -1, 10, -2, 8, -3, 5, -6, 7, -4, 3, -8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)+t(2))\left(2t(2)t(1)^{2}-t(1)^{2}+2t(2)^{2}t(1)-4t(2)t(1)+2t(1)-t(2)^{2}+2t(2)\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {6}{q^{7/2}}}+{\frac {7}{q^{9/2}}}-{\frac {9}{q^{11/2}}}+{\frac {9}{q^{13/2}}}-{\frac {8}{q^{15/2}}}+{\frac {6}{q^{17/2}}}-{\frac {4}{q^{19/2}}}+{\frac {2}{q^{21/2}}}-{\frac {1}{q^{23/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$za^{11}-z^{3}a^{9}+za^{9}+a^{9}z^{-1}-3z^{3}a^{7}-3za^{7}-a^{7}z^{-1}-3z^{3}a^{5}-3za^{5}-z^{3}a^{3}$ (db)
Kauffman polynomial	$a^{13}z^{7}-5a^{13}z^{5}+8a^{13}z^{3}-4a^{13}z+2a^{12}z^{8}-9a^{12}z^{6}+12a^{12}z^{4}-4a^{12}z^{2}+a^{11}z^{9}-11a^{11}z^{5}+15a^{11}z^{3}-3a^{11}z+5a^{10}z^{8}-15a^{10}z^{6}+9a^{10}z^{4}+a^{10}z^{2}+a^{9}z^{9}+5a^{9}z^{7}-21a^{9}z^{5}+18a^{9}z^{3}-7a^{9}z+a^{9}z^{-1}+3a^{8}z^{8}+a^{8}z^{6}-15a^{8}z^{4}+8a^{8}z^{2}-a^{8}+6a^{7}z^{7}-9a^{7}z^{5}+4a^{7}z^{3}-5a^{7}z+a^{7}z^{-1}+7a^{6}z^{6}-9a^{6}z^{4}+3a^{6}z^{2}+6a^{5}z^{5}-6a^{5}z^{3}+3a^{5}z+3a^{4}z^{4}+a^{3}z^{3}$ (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 -2                     1 1 -4                   3 1 -2 -6                 3     3 -8               4 3     -1 -10             5 3       2 -12           4 4         0 -14         4 5           -1 -16       2 4             2 -18     2 4               -2 -20   1 3                 2 -22   1                   -1 -24 1                     1

Integral Khovanov Homology (db, data source)

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

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.

L10a100 L10a102