L11n82

L11n81

L11n83

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n82 at Knotilus!

Link Presentations

[edit Notes on L11n82's Link Presentations]

Planar diagram presentation	X6172 X20,7,21,8 X4,21,1,22 X5,14,6,15 X3,10,4,11 X11,16,12,17 X15,12,16,13 X13,22,14,5 X18,9,19,10 X17,2,18,3 X8,19,9,20
Gauss code	{1, 10, -5, -3}, {-4, -1, 2, -11, 9, 5, -6, 7, -8, 4, -7, 6, -10, -9, 11, -2, 3, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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