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
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{4 u^2 v^2-4 u^2 v+u^2-4 u v^2+9 u v-4 u+v^2-4 v+4}{u v}$ (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	$z^3 a^{11}+z a^{11}-z^5 a^9-z^3 a^9+z a^9-2 z^5 a^7-5 z^3 a^7-2 z a^7+a^7 z^{-1} -z^5 a^5-3 z^3 a^5-3 z a^5-a^5 z^{-1}$ (db)
Kauffman polynomial	$-z^5 a^{15}+2 z^3 a^{15}-z a^{15}-3 z^6 a^{14}+6 z^4 a^{14}-3 z^2 a^{14}-4 z^7 a^{13}+6 z^5 a^{13}-z^3 a^{13}-z a^{13}-3 z^8 a^{12}+z^6 a^{12}+3 z^4 a^{12}-z^9 a^{11}-5 z^7 a^{11}+6 z^5 a^{11}+3 z^3 a^{11}-3 z a^{11}-5 z^8 a^{10}+3 z^6 a^{10}+4 z^4 a^{10}-3 z^2 a^{10}-z^9 a^9-4 z^7 a^9+4 z^5 a^9+2 z^3 a^9-z a^9-2 z^8 a^8-3 z^6 a^8+10 z^4 a^8-6 z^2 a^8-3 z^7 a^7+4 z^5 a^7-z^3 a^7-z a^7+a^7 z^{-1} -2 z^6 a^6+3 z^4 a^6-a^6-z^5 a^5+3 z^3 a^5-3 z a^5+a^5 z^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials $t^rq^j$ 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)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-10$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-7$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-6$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=-5$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

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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