L11n180

L11n179

L11n181

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n180 at Knotilus!

Link Presentations

[edit Notes on L11n180's Link Presentations]

Planar diagram presentation	X8192 X3,10,4,11 X5,14,6,15 X16,8,17,7 X22,18,7,17 X15,13,16,12 X9,20,10,21 X11,19,12,18 X13,6,14,1 X19,4,20,5 X2,21,3,22
Gauss code	{1, -11, -2, 10, -3, 9}, {4, -1, -7, 2, -8, 6, -9, 3, -6, -4, 5, 8, -10, 7, 11, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{u^2 v^2-u^2 v-u^2-u v^2+u v-u-v^2-v+1}{u v}$ (db)
Jones polynomial	$-\frac{1}{q^{7/2}}-q^{5/2}+\frac{1}{q^{5/2}}+2 q^{3/2}-\frac{2}{q^{3/2}}-\frac{1}{q^{13/2}}-2 \sqrt{q}+\frac{2}{\sqrt{q}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^7 z^{-1} -a^5 z-a^5 z^{-1} -a^3 z^3-3 a^3 z+a z^5+4 a z^3-z^3 a^{-1} +3 a z-2 z a^{-1}$ (db)
Kauffman polynomial	$a^7 z^5-5 a^7 z^3+5 a^7 z-a^7 z^{-1} -2 a^6 z^2+a^6-3 a^5 z^3+3 a^5 z-a^5 z^{-1} +a^4 z^8-6 a^4 z^6+8 a^4 z^4-3 a^4 z^2+a^3 z^9-5 a^3 z^7+4 a^3 z^5+2 a^3 z^3-2 a^3 z+3 a^2 z^8-17 a^2 z^6+24 a^2 z^4-8 a^2 z^2+a z^9-4 a z^7+z^7 a^{-1} -5 z^5 a^{-1} +6 a z^3+6 z^3 a^{-1} -2 a z-2 z a^{-1} +2 z^8-11 z^6+16 z^4-7 z^2$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 6                     1 1 4                   1   -1 2                 1 1   0 0             1 2 1     0 -2           1 2 1       0 -4           1 2         1 -6       1 2 1           0 -8         1             1 -10     1 1               0 -12 1                     1 -14 1                     1

Integral Khovanov Homology (db, data source)

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

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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