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 $u$, $v$, $w$, ...)	$\frac{2 u^2 v^3-5 u^2 v^2+2 u^2 v+u v^4-6 u v^3+9 u v^2-6 u v+u+2 v^3-5 v^2+2 v}{u v^2}$ (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	$z a^9+a^9 z^{-1} -3 z^3 a^7-5 z a^7-2 a^7 z^{-1} +2 z^5 a^5+5 z^3 a^5+5 z a^5+2 a^5 z^{-1} +z^5 a^3-3 z a^3-a^3 z^{-1} -z^3 a-z a$ (db)
Kauffman polynomial	$a^{11} z^5-2 a^{11} z^3+a^{11} z+3 a^{10} z^6-5 a^{10} z^4+a^{10} z^2+5 a^9 z^7-9 a^9 z^5+6 a^9 z^3-5 a^9 z+a^9 z^{-1} +5 a^8 z^8-8 a^8 z^6+5 a^8 z^4-2 a^8 z^2+2 a^7 z^9+6 a^7 z^7-22 a^7 z^5+26 a^7 z^3-12 a^7 z+2 a^7 z^{-1} +10 a^6 z^8-22 a^6 z^6+20 a^6 z^4-6 a^6 z^2+a^6+2 a^5 z^9+6 a^5 z^7-22 a^5 z^5+26 a^5 z^3-12 a^5 z+2 a^5 z^{-1} +5 a^4 z^8-8 a^4 z^6+5 a^4 z^4-2 a^4 z^2+5 a^3 z^7-9 a^3 z^5+6 a^3 z^3-5 a^3 z+a^3 z^{-1} +3 a^2 z^6-5 a^2 z^4+a^2 z^2+a z^5-2 a z^3+a z$ (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   \ -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)

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

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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