L11n150

L11n149

L11n151

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n150 at Knotilus!

Link Presentations

[edit Notes on L11n150's Link Presentations]

Planar diagram presentation	X8192 X18,9,19,10 X6718 X22,19,7,20 X12,5,13,6 X3,10,4,11 X15,5,16,4 X11,16,12,17 X20,13,21,14 X14,21,15,22 X2,18,3,17
Gauss code	{1, -11, -6, 7, 5, -3}, {3, -1, 2, 6, -8, -5, 9, -10, -7, 8, 11, -2, 4, -9, 10, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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