L11a203

L11a202

L11a204

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a203 at Knotilus!

Link Presentations

[edit Notes on L11a203's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X6718 X18,11,19,12 X16,6,17,5 X4,18,5,17 X22,15,7,16 X12,21,13,22 X20,13,21,14 X14,19,15,20
Gauss code	{1, -2, 3, -7, 6, -4}, {4, -1, 2, -3, 5, -9, 10, -11, 8, -6, 7, -5, 11, -10, 9, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2u^{2}v^{4}-3u^{2}v^{3}+3u^{2}v^{2}-2u^{2}v-2uv^{4}+3uv^{3}-3uv^{2}+3uv-2u-2v^{3}+3v^{2}-3v+2}{uv^{2}}}$ (db)
Jones polynomial	$-{\frac {1}{\sqrt {q}}}+{\frac {2}{q^{3/2}}}-{\frac {5}{q^{5/2}}}+{\frac {6}{q^{7/2}}}-{\frac {9}{q^{9/2}}}+{\frac {10}{q^{11/2}}}-{\frac {10}{q^{13/2}}}+{\frac {9}{q^{15/2}}}-{\frac {7}{q^{17/2}}}+{\frac {4}{q^{19/2}}}-{\frac {2}{q^{21/2}}}+{\frac {1}{q^{23/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-z^{5}a^{9}-4z^{3}a^{9}-4za^{9}-a^{9}z^{-1}+z^{7}a^{7}+5z^{5}a^{7}+9z^{3}a^{7}+8za^{7}+2a^{7}z^{-1}+z^{7}a^{5}+4z^{5}a^{5}+3z^{3}a^{5}-za^{5}-z^{5}a^{3}-4z^{3}a^{3}-4za^{3}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$-z^{4}a^{14}+2z^{2}a^{14}-2z^{5}a^{13}+3z^{3}a^{13}-3z^{6}a^{12}+5z^{4}a^{12}-3z^{2}a^{12}-4z^{7}a^{11}+10z^{5}a^{11}-12z^{3}a^{11}+2za^{11}-4z^{8}a^{10}+12z^{6}a^{10}-18z^{4}a^{10}+9z^{2}a^{10}-2a^{10}-3z^{9}a^{9}+9z^{7}a^{9}-13z^{5}a^{9}+9z^{3}a^{9}-4za^{9}+a^{9}z^{-1}-z^{10}a^{8}-2z^{8}a^{8}+17z^{6}a^{8}-31z^{4}a^{8}+25z^{2}a^{8}-5a^{8}-5z^{9}a^{7}+20z^{7}a^{7}-31z^{5}a^{7}+29z^{3}a^{7}-12za^{7}+2a^{7}z^{-1}-z^{10}a^{6}+10z^{6}a^{6}-14z^{4}a^{6}+10z^{2}a^{6}-3a^{6}-2z^{9}a^{5}+6z^{7}a^{5}-z^{5}a^{5}-3z^{3}a^{5}-za^{5}-2z^{8}a^{4}+8z^{6}a^{4}-7z^{4}a^{4}-z^{2}a^{4}+a^{4}-z^{7}a^{3}+5z^{5}a^{3}-8z^{3}a^{3}+5za^{3}-a^{3}z^{-1}$ (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                     1   -1 -4                   4 1   3 -6                 3 2     -1 -8               6 3       3 -10             5 4         -1 -12           5 5           0 -14         4 5             1 -16       3 5               -2 -18     1 4                 3 -20   1 3                   -2 -22   1                     1 -24 1                       -1

Integral Khovanov Homology (db, data source)

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

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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