L10a81

L10a80

L10a82

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a81 at Knotilus!

Link Presentations

[edit Notes on L10a81's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X14,19,15,20 X16,5,17,6 X18,10,19,9 X4,15,5,16 X10,18,11,17 X20,13,7,14 X2738 X6,11,1,12
Gauss code	{1, -9, 2, -6, 4, -10}, {9, -1, 5, -7, 10, -2, 8, -3, 6, -4, 7, -5, 3, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2u^{2}v^{3}-5u^{2}v^{2}+2u^{2}v+uv^{4}-6uv^{3}+9uv^{2}-6uv+u+2v^{3}-5v^{2}+2v}{uv^{2}}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {7}{q^{3/2}}}+{\frac {10}{q^{5/2}}}-{\frac {13}{q^{7/2}}}+{\frac {14}{q^{9/2}}}-{\frac {13}{q^{11/2}}}+{\frac {10}{q^{13/2}}}-{\frac {7}{q^{15/2}}}+{\frac {3}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$za^{9}+a^{9}z^{-1}-3z^{3}a^{7}-5za^{7}-2a^{7}z^{-1}+2z^{5}a^{5}+5z^{3}a^{5}+5za^{5}+2a^{5}z^{-1}+z^{5}a^{3}-3za^{3}-a^{3}z^{-1}-z^{3}a-za$ (db)
Kauffman polynomial	$a^{11}z^{5}-2a^{11}z^{3}+a^{11}z+3a^{10}z^{6}-5a^{10}z^{4}+a^{10}z^{2}+5a^{9}z^{7}-9a^{9}z^{5}+6a^{9}z^{3}-5a^{9}z+a^{9}z^{-1}+5a^{8}z^{8}-8a^{8}z^{6}+5a^{8}z^{4}-2a^{8}z^{2}+2a^{7}z^{9}+6a^{7}z^{7}-22a^{7}z^{5}+26a^{7}z^{3}-12a^{7}z+2a^{7}z^{-1}+10a^{6}z^{8}-22a^{6}z^{6}+20a^{6}z^{4}-6a^{6}z^{2}+a^{6}+2a^{5}z^{9}+6a^{5}z^{7}-22a^{5}z^{5}+26a^{5}z^{3}-12a^{5}z+2a^{5}z^{-1}+5a^{4}z^{8}-8a^{4}z^{6}+5a^{4}z^{4}-2a^{4}z^{2}+5a^{3}z^{7}-9a^{3}z^{5}+6a^{3}z^{3}-5a^{3}z+a^{3}z^{-1}+3a^{2}z^{6}-5a^{2}z^{4}+a^{2}z^{2}+az^{5}-2az^{3}+az$ (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 1 2 χ 2                     1 1 0                   2   -2 -2                 5 1   4 -4               6 3     -3 -6             7 4       3 -8           7 6         -1 -10         6 7           -1 -12       4 7             3 -14     3 6               -3 -16   1 5                 4 -18   2                   -2 -20 1                     1

Integral Khovanov Homology (db, data source)

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

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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