L11a212

L11a211

L11a213

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a212 at Knotilus!

Link Presentations

[edit Notes on L11a212's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X14,8,15,7 X6,13,1,14 X20,17,21,18 X16,5,17,6 X18,11,19,12 X12,19,13,20 X22,16,7,15 X4,21,5,22
Gauss code	{1, -2, 3, -11, 7, -5}, {4, -1, 2, -3, 8, -9, 5, -4, 10, -7, 6, -8, 9, -6, 11, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2u^{2}v^{4}-6u^{2}v^{3}+5u^{2}v^{2}-2u^{2}v-2uv^{4}+7uv^{3}-11uv^{2}+7uv-2u-2v^{3}+5v^{2}-6v+2}{uv^{2}}}$ (db)
Jones polynomial	$-{\frac {16}{q^{9/2}}}+{\frac {11}{q^{7/2}}}-{\frac {7}{q^{5/2}}}+{\frac {3}{q^{3/2}}}+{\frac {1}{q^{23/2}}}-{\frac {4}{q^{21/2}}}+{\frac {8}{q^{19/2}}}-{\frac {13}{q^{17/2}}}+{\frac {17}{q^{15/2}}}-{\frac {19}{q^{13/2}}}+{\frac {18}{q^{11/2}}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{9}\left(-z^{5}\right)-2a^{9}z^{3}+a^{7}z^{7}+3a^{7}z^{5}+2a^{7}z^{3}+a^{7}z+a^{7}z^{-1}+a^{5}z^{7}+3a^{5}z^{5}+a^{5}z^{3}-2a^{5}z-a^{5}z^{-1}-a^{3}z^{5}-3a^{3}z^{3}-2a^{3}z$ (db)
Kauffman polynomial	$-z^{4}a^{14}-4z^{5}a^{13}+2z^{3}a^{13}-8z^{6}a^{12}+7z^{4}a^{12}-z^{2}a^{12}-11z^{7}a^{11}+14z^{5}a^{11}-5z^{3}a^{11}-11z^{8}a^{10}+17z^{6}a^{10}-7z^{4}a^{10}-7z^{9}a^{9}+6z^{7}a^{9}+9z^{5}a^{9}-8z^{3}a^{9}+2za^{9}-2z^{10}a^{8}-11z^{8}a^{8}+39z^{6}a^{8}-31z^{4}a^{8}+7z^{2}a^{8}-11z^{9}a^{7}+30z^{7}a^{7}-22z^{5}a^{7}+7z^{3}a^{7}-3za^{7}+a^{7}z^{-1}-2z^{10}a^{6}-3z^{8}a^{6}+25z^{6}a^{6}-28z^{4}a^{6}+10z^{2}a^{6}-a^{6}-4z^{9}a^{5}+12z^{7}a^{5}-9z^{5}a^{5}+3z^{3}a^{5}-3za^{5}+a^{5}z^{-1}-3z^{8}a^{4}+11z^{6}a^{4}-12z^{4}a^{4}+4z^{2}a^{4}-z^{7}a^{3}+4z^{5}a^{3}-5z^{3}a^{3}+2za^{3}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 0                       1 1 -2                     2   -2 -4                   5 1   4 -6                 7 3     -4 -8               9 4       5 -10             9 7         -2 -12           10 9           1 -14         7 9             2 -16       6 10               -4 -18     3 8                 5 -20   1 5                   -4 -22   3                     3 -24 1                       -1

Integral Khovanov Homology (db, data source)

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

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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