L11n64

L11n63

L11n65

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n64 at Knotilus!

Link Presentations

[edit Notes on L11n64's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,8,15,7 X15,20,16,21 X11,18,12,19 X19,12,20,13 X17,22,18,5 X21,16,22,17 X8,14,9,13 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, -5, 6, 9, -3, -4, 8, -7, 5, -6, 4, -8, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{u v^5-2 u v^4+u v^2-u-v^5+v^3-2 v+1}{\sqrt{u} v^{5/2}}$ (db)
Jones polynomial	$-\frac{1}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{3}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{1}{q^{11/2}}-\frac{2}{q^{13/2}}+\frac{1}{q^{17/2}}-\frac{1}{q^{19/2}}+\frac{1}{q^{21/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-a^{11} z^{-1} +3 a^9 z+2 a^9 z^{-1} -a^7 z^5-5 a^7 z^3-4 a^7 z+a^5 z^7+5 a^5 z^5+6 a^5 z^3+2 a^5 z-a^5 z^{-1} -a^3 z^5-4 a^3 z^3-3 a^3 z$ (db)
Kauffman polynomial	$-z^6 a^{12}+5 z^4 a^{12}-6 z^2 a^{12}+2 a^{12}-z^7 a^{11}+5 z^5 a^{11}-5 z^3 a^{11}+2 z a^{11}-a^{11} z^{-1} -z^6 a^{10}+8 z^4 a^{10}-12 z^2 a^{10}+5 a^{10}+2 z^5 a^9-3 z^3 a^9+3 z a^9-2 a^9 z^{-1} -z^8 a^8+6 z^6 a^8-6 z^4 a^8-z^2 a^8+3 a^8-z^9 a^7+5 z^7 a^7-5 z^5 a^7+3 z^3 a^7-3 z a^7-3 z^8 a^6+16 z^6 a^6-22 z^4 a^6+10 z^2 a^6-a^6-z^9 a^5+3 z^7 a^5+3 z^5 a^5-6 z^3 a^5-z a^5+a^5 z^{-1} -2 z^8 a^4+10 z^6 a^4-13 z^4 a^4+5 z^2 a^4-z^7 a^3+5 z^5 a^3-7 z^3 a^3+3 z a^3$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 0                       1 1 -2                     1   -1 -4                   2 1   1 -6                 2 2     0 -8             1 2 1       0 -10           1 1 2         2 -12         1 4 2           1 -14       1   1             2 -16       1 2               -1 -18   1 1                   0 -20                         0 -22 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $i=-2$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{4}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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