L11n287

L11n286

L11n288

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n287 at Knotilus!

Link Presentations

[edit Notes on L11n287's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X8493 X2,14,3,13 X14,7,15,8 X4,15,1,16 X21,18,22,19 X9,21,10,20 X19,5,20,10 X11,16,12,17 X17,22,18,11
Gauss code	{1, -4, 3, -6}, {-2, -1, 5, -3, -8, 9}, {-10, 2, 4, -5, 6, 10, -11, 7, -9, 8, -7, 11}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(u-1) (v-1) (w-1)}{\sqrt{u} \sqrt{v} \sqrt{w}}$ (db)
Jones polynomial	$q^2+2- q^{-1} +3 q^{-2} -2 q^{-3} +2 q^{-4} -2 q^{-5} +2 q^{-6} - q^{-7}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^6 \left(-z^2\right)-a^6+a^4 z^4+3 a^4 z^2+2 a^4-a^2 z^2+a^2 z^{-2} + a^{-2} z^{-2} + a^{-2} -z^2-2 z^{-2} -2$ (db)
Kauffman polynomial	$a^5 z^9+a^3 z^9+2 a^6 z^8+3 a^4 z^8+a^2 z^8+a^7 z^7-4 a^5 z^7-5 a^3 z^7-11 a^6 z^6-17 a^4 z^6-6 a^2 z^6-5 a^7 z^5+6 a^3 z^5+a z^5+16 a^6 z^4+28 a^4 z^4+12 a^2 z^4+6 a^7 z^3+8 a^5 z^3-2 a^3 z^3-4 a z^3-7 a^6 z^2-18 a^4 z^2-10 a^2 z^2+z^2 a^{-2} +2 z^2-2 a^7 z-4 a^5 z+4 a z+2 z a^{-1} +2 a^6+4 a^4-2 a^{-2} -3-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 5                   1 1 3                   1 1 1               3 1   2 -1             2 4 1   1 -3           2 2 2     2 -5         2 3         1 -7       1 2 1         0 -9     1 2 1           0 -11   1 1               0 -13   1                 1 -15 1                   -1

Integral Khovanov Homology (db, data source)

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

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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