L11n212

L11n211

L11n213

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n212 at Knotilus!

Link Presentations

[edit Notes on L11n212's Link Presentations]

Planar diagram presentation	X10,1,11,2 X13,21,14,20 X3,12,4,13 X2,19,3,20 X14,5,15,6 X16,7,17,8 X8,9,1,10 X18,12,19,11 X6,15,7,16 X22,18,9,17 X21,4,22,5
Gauss code	{1, -4, -3, 11, 5, -9, 6, -7}, {7, -1, 8, 3, -2, -5, 9, -6, 10, -8, 4, 2, -11, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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