L11n127

L11n126

L11n128

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n127 at Knotilus!

Link Presentations

[edit Notes on L11n127's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X22,10,7,9 X2738 X4,15,5,16 X5,13,6,12 X11,16,12,17 X17,6,18,1 X14,20,15,19 X20,14,21,13 X18,21,19,22
Gauss code	{1, -4, 2, -5, -6, 8}, {4, -1, 3, -2, -7, 6, 10, -9, 5, 7, -8, -11, 9, -10, 11, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{u^2 v^2-2 u^2 v+u^2-u v^2+u v-u+v^2-2 v+1}{u v}$ (db)
Jones polynomial	$-q^{5/2}+2 q^{3/2}-3 \sqrt{q}+\frac{3}{\sqrt{q}}-\frac{4}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{3}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^5 z+a^5 z^{-1} -2 a^3 z^3-5 a^3 z-2 a^3 z^{-1} +a z^5+4 a z^3-z^3 a^{-1} +5 a z+2 a z^{-1} -2 z a^{-1} - a^{-1} z^{-1}$ (db)
Kauffman polynomial	$-a^3 z^9-a z^9-2 a^4 z^8-4 a^2 z^8-2 z^8-a^5 z^7+2 a^3 z^7+2 a z^7-z^7 a^{-1} +10 a^4 z^6+20 a^2 z^6+10 z^6+5 a^5 z^5+9 a^3 z^5+9 a z^5+5 z^5 a^{-1} -13 a^4 z^4-26 a^2 z^4-13 z^4-7 a^5 z^3-20 a^3 z^3-20 a z^3-7 z^3 a^{-1} +5 a^4 z^2+10 a^2 z^2+5 z^2+4 a^5 z+12 a^3 z+12 a z+4 z a^{-1} -a^2-a^5 z^{-1} -2 a^3 z^{-1} -2 a z^{-1} - a^{-1} z^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 χ 6                 1 1 4               1   -1 2             2 1   1 0           2 2     0 -2         2 2 1     1 -4       2 3         1 -6     1 2 1         0 -8   1 2             1 -10   1               -1 -12 1                 1

Integral Khovanov Homology (db, data source)

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

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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