L10a10

L10a9

L10a11

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a10 at Knotilus!

Link Presentations

[edit Notes on L10a10's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,8,13,7 X18,16,19,15 X16,9,17,10 X8,17,9,18 X20,14,5,13 X14,20,15,19 X2536 X4,12,1,11
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -6, 5, -2, 10, -3, 7, -8, 4, -5, 6, -4, 8, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{2 (t(1)-1) (t(2)-1)^3}{\sqrt{t(1)} t(2)^{3/2}}$ (db)
Jones polynomial	$-q^{13/2}+3 q^{11/2}-6 q^{9/2}+8 q^{7/2}-10 q^{5/2}+11 q^{3/2}-10 \sqrt{q}+\frac{7}{\sqrt{q}}-\frac{5}{q^{3/2}}+\frac{2}{q^{5/2}}-\frac{1}{q^{7/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^3 a^{-5} -z a^{-5} - a^{-5} z^{-1} +z^5 a^{-3} +2 z^3 a^{-3} +a^3 z+3 z a^{-3} +a^3 z^{-1} +2 a^{-3} z^{-1} +z^5 a^{-1} -2 a z^3+z^3 a^{-1} -3 a z-a z^{-1} - a^{-1} z^{-1}$ (db)
Kauffman polynomial	$-z^9 a^{-1} -z^9 a^{-3} -6 z^8 a^{-2} -4 z^8 a^{-4} -2 z^8-2 a z^7-3 z^7 a^{-1} -6 z^7 a^{-3} -5 z^7 a^{-5} -2 a^2 z^6+15 z^6 a^{-2} +9 z^6 a^{-4} -3 z^6 a^{-6} +z^6-a^3 z^5+8 z^5 a^{-1} +21 z^5 a^{-3} +13 z^5 a^{-5} -z^5 a^{-7} +4 a^2 z^4-20 z^4 a^{-2} -9 z^4 a^{-4} +6 z^4 a^{-6} -z^4+3 a^3 z^3+7 a z^3-10 z^3 a^{-1} -27 z^3 a^{-3} -11 z^3 a^{-5} +2 z^3 a^{-7} -a^2 z^2+11 z^2 a^{-2} +4 z^2 a^{-4} -z^2 a^{-6} +5 z^2-3 a^3 z-5 a z+6 z a^{-1} +13 z a^{-3} +5 z a^{-5} -a^2-3 a^{-2} - a^{-4} -2+a^3 z^{-1} +a z^{-1} - a^{-1} z^{-1} -2 a^{-3} z^{-1} - a^{-5} z^{-1}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                     1 1 12                   2   -2 10                 4 1   3 8               4 2     -2 6             6 4       2 4           5 4         -1 2         5 6           -1 0       4 7             3 -2     1 3               -2 -4   1 4                 3 -6   1                   -1 -8 1                     1

Integral Khovanov Homology (db, data source)

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

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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