L11n229

L11n228

L11n230

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n229 at Knotilus!

Link Presentations

[edit Notes on L11n229's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X20,5,21,6 X7,14,8,15 X13,16,14,17 X15,8,16,1 X22,17,9,18 X18,21,19,22 X6,9,7,10 X4,19,5,20
Gauss code	{1, -2, 3, -11, 4, -10, -5, 7}, {10, -1, 2, -3, -6, 5, -7, 6, 8, -9, 11, -4, 9, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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.

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