L10a87

L10a86

L10a88

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a87 at Knotilus!

Other depiction

Mongolian ornament

Link Presentations

[edit Notes on L10a87's Link Presentations]

Planar diagram presentation	X8192 X20,9,7,10 X6718 X2,11,3,12 X12,3,13,4 X16,5,17,6 X18,13,19,14 X14,17,15,18 X4,15,5,16 X10,19,11,20
Gauss code	{1, -4, 5, -9, 6, -3}, {3, -1, 2, -10, 4, -5, 7, -8, 9, -6, 8, -7, 10, -2}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {4u^{2}v^{2}-4u^{2}v+u^{2}-4uv^{2}+9uv-4u+v^{2}-4v+4}{uv}}$ (db)
Jones polynomial	$-{\frac {6}{q^{9/2}}}+{\frac {2}{q^{7/2}}}-{\frac {1}{q^{5/2}}}-{\frac {1}{q^{25/2}}}+{\frac {3}{q^{23/2}}}-{\frac {6}{q^{21/2}}}+{\frac {9}{q^{19/2}}}-{\frac {11}{q^{17/2}}}+{\frac {12}{q^{15/2}}}-{\frac {11}{q^{13/2}}}+{\frac {8}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$z^{3}a^{11}+za^{11}-z^{5}a^{9}-z^{3}a^{9}+za^{9}-2z^{5}a^{7}-5z^{3}a^{7}-2za^{7}+a^{7}z^{-1}-z^{5}a^{5}-3z^{3}a^{5}-3za^{5}-a^{5}z^{-1}$ (db)
Kauffman polynomial	$-z^{5}a^{15}+2z^{3}a^{15}-za^{15}-3z^{6}a^{14}+6z^{4}a^{14}-3z^{2}a^{14}-4z^{7}a^{13}+6z^{5}a^{13}-z^{3}a^{13}-za^{13}-3z^{8}a^{12}+z^{6}a^{12}+3z^{4}a^{12}-z^{9}a^{11}-5z^{7}a^{11}+6z^{5}a^{11}+3z^{3}a^{11}-3za^{11}-5z^{8}a^{10}+3z^{6}a^{10}+4z^{4}a^{10}-3z^{2}a^{10}-z^{9}a^{9}-4z^{7}a^{9}+4z^{5}a^{9}+2z^{3}a^{9}-za^{9}-2z^{8}a^{8}-3z^{6}a^{8}+10z^{4}a^{8}-6z^{2}a^{8}-3z^{7}a^{7}+4z^{5}a^{7}-z^{3}a^{7}-za^{7}+a^{7}z^{-1}-2z^{6}a^{6}+3z^{4}a^{6}-a^{6}-z^{5}a^{5}+3z^{3}a^{5}-3za^{5}+a^{5}z^{-1}$ (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   \ -10 -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             7 4       3 -14           5 4         -1 -16         6 7           -1 -18       4 6             2 -20     2 5               -3 -22   1 4                 3 -24   2                   -2 -26 1                     1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-6$ $i=-4$ $r=-10$ ${\mathbb {Z} }$ $r=-9$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-7$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-6$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=-5$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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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