L11n85

L11n84

L11n86

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n85 at Knotilus!

Link Presentations

[edit Notes on L11n85's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X14,8,15,7 X9,18,10,19 X22,19,5,20 X20,15,21,16 X16,21,17,22 X17,8,18,9 X10,14,11,13 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, 8, -4, -9, 11, -2, 9, -3, 6, -7, -8, 4, 5, -6, 7, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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