L11n187

L11n186

L11n188

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n187 at Knotilus!

Link Presentations

[edit Notes on L11n187's Link Presentations]

Planar diagram presentation	X8192 X12,4,13,3 X22,12,7,11 X15,20,16,21 X18,10,19,9 X10,20,11,19 X21,14,22,15 X16,6,17,5 X2738 X4,14,5,13 X6,18,1,17
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 5, -6, 3, -2, 10, 7, -4, -8, 11, -5, 6, 4, -7, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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