L11n180

L11n179

L11n181

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n180 at Knotilus!

Link Presentations

[edit Notes on L11n180's Link Presentations]

Planar diagram presentation	X8192 X3,10,4,11 X5,14,6,15 X16,8,17,7 X22,18,7,17 X15,13,16,12 X9,20,10,21 X11,19,12,18 X13,6,14,1 X19,4,20,5 X2,21,3,22
Gauss code	{1, -11, -2, 10, -3, 9}, {4, -1, -7, 2, -8, 6, -9, 3, -6, -4, 5, 8, -10, 7, 11, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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