L11n420

L11n419

L11n421

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n420 at Knotilus!

Link Presentations

[edit Notes on L11n420's Link Presentations]

Planar diagram presentation	X8192 X7,16,8,17 X3,10,4,11 X17,2,18,3 X18,9,19,10 X11,20,12,21 X14,6,15,5 X22,15,13,16 X6,14,1,13 X4,19,5,20 X21,12,22,7
Gauss code	{1, 4, -3, -10, 7, -9}, {-2, -1, 5, 3, -6, 11}, {9, -7, 8, 2, -4, -5, 10, 6, -11, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{u^2 v^2 w^3-u^2 v^2 w^2-u v^2 w^3+u v w^2-u v w+u+w-1}{u v w^{3/2}}$ (db)
Jones polynomial	$q^{-7} +2 q^{-5} - q^{-4} +2 q^{-3} - q^{-2} + q^{-1}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$a^{10} \left(-z^2\right)-a^{10}+a^8 z^6+6 a^8 z^4+9 a^8 z^2+a^8 z^{-2} +4 a^8-a^6 z^8-7 a^6 z^6-16 a^6 z^4-16 a^6 z^2-2 a^6 z^{-2} -8 a^6+a^4 z^6+6 a^4 z^4+10 a^4 z^2+a^4 z^{-2} +5 a^4$ (db)
Kauffman polynomial	$a^{11} (-z)-2 a^{10} z^2+a^{10}+a^9 z^7-6 a^9 z^5+8 a^9 z^3-3 a^9 z+2 a^8 z^8-13 a^8 z^6+25 a^8 z^4-20 a^8 z^2-a^8 z^{-2} +8 a^8+a^7 z^9-5 a^7 z^7+3 a^7 z^5+8 a^7 z^3-8 a^7 z+2 a^7 z^{-1} +3 a^6 z^8-20 a^6 z^6+41 a^6 z^4-33 a^6 z^2-2 a^6 z^{-2} +12 a^6+a^5 z^9-6 a^5 z^7+9 a^5 z^5-6 a^5 z+2 a^5 z^{-1} +a^4 z^8-7 a^4 z^6+16 a^4 z^4-15 a^4 z^2-a^4 z^{-2} +6 a^4$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 χ -1                 1 1 -3                   0 -5             2 1   1 -7         1 1 1     1 -9         2 1       1 -11     2 1 1         2 -13     2 1           1 -15 1 1 1             1 -17 1 1               0

Integral Khovanov Homology (db, data source)

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

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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