L11n210

L11n209

L11n211

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n210 at Knotilus!

Link Presentations

[edit Notes on L11n210's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X5,14,6,15 X22,18,9,17 X4,19,5,20 X21,6,22,7 X16,7,17,8 X8,9,1,10 X18,14,19,13 X15,21,16,20
Gauss code	{1, -2, 3, -6, -4, 7, 8, -9}, {9, -1, 2, -3, 10, 4, -11, -8, 5, -10, 6, 11, -7, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{3}v^{2}+u^{2}v^{5}-u^{2}v^{4}+u^{2}v^{3}-u^{2}v^{2}+u^{2}v+uv^{4}-uv^{3}+uv^{2}-uv+u+v^{3}}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	${\frac {1}{\sqrt {q}}}-{\frac {2}{q^{3/2}}}+{\frac {2}{q^{5/2}}}-{\frac {4}{q^{7/2}}}+{\frac {4}{q^{9/2}}}-{\frac {4}{q^{11/2}}}+{\frac {3}{q^{13/2}}}-{\frac {2}{q^{15/2}}}+{\frac {1}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z^{5}a^{7}+5z^{3}a^{7}+7za^{7}+2a^{7}z^{-1}-z^{7}a^{5}-7z^{5}a^{5}-17z^{3}a^{5}-15za^{5}-3a^{5}z^{-1}+z^{5}a^{3}+4z^{3}a^{3}+4za^{3}+a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{11}z^{5}-4a^{11}z^{3}+3a^{11}z+a^{10}z^{6}-3a^{10}z^{4}+a^{10}z^{2}+a^{9}z^{7}-3a^{9}z^{5}+2a^{9}z^{3}-a^{9}z+a^{8}z^{8}-4a^{8}z^{6}+6a^{8}z^{4}-3a^{8}z^{2}+a^{7}z^{9}-6a^{7}z^{7}+15a^{7}z^{5}-16a^{7}z^{3}+10a^{7}z-2a^{7}z^{-1}+2a^{6}z^{8}-11a^{6}z^{6}+23a^{6}z^{4}-15a^{6}z^{2}+3a^{6}+a^{5}z^{9}-7a^{5}z^{7}+21a^{5}z^{5}-28a^{5}z^{3}+18a^{5}z-3a^{5}z^{-1}+a^{4}z^{8}-6a^{4}z^{6}+15a^{4}z^{4}-14a^{4}z^{2}+3a^{4}+2a^{3}z^{5}-6a^{3}z^{3}+4a^{3}z-a^{3}z^{-1}+a^{2}z^{4}-3a^{2}z^{2}+a^{2}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 χ 0                   1 -1 -2                 1   1 -4               2 2   0 -6             2       2 -8           2 2       0 -10         2 2         0 -12       1 2           1 -14     1 2             -1 -16     1               1 -18 1 1                 0 -20 1                   1

Integral Khovanov Homology (db, data source)

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

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.

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