L11n166

L11n165

L11n167

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Link Presentations

[edit Notes on L11n166's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X11,19,12,18 X19,7,20,22 X15,21,16,20 X21,17,22,16 X17,13,18,12 X6718 X4,13,5,14
Gauss code	{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, -5, 9, 11, -4, -7, 8, -9, 5, -6, 7, -8, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{t(1)^2 t(2)^4+t(1) t(2)^4-t(1)^2 t(2)^3-t(1) t(2)^3+t(2)^3+t(1) t(2)^2+t(1)^2 t(2)-t(1) t(2)-t(2)+t(1)+1}{t(1) t(2)^2}$ (db)
Jones polynomial	$q^{7/2}-q^{5/2}+q^{3/2}-\sqrt{q}-\frac{1}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{2}{q^{5/2}}+\frac{2}{q^{7/2}}-\frac{2}{q^{9/2}}+\frac{1}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a z^7-7 a z^5+z^5 a^{-1} +a^5 z^3+a^3 z^3-15 a z^3+5 z^3 a^{-1} +2 a^5 z+a^3 z-12 a z+6 z a^{-1} +a^5 z^{-1} -2 a z^{-1} + a^{-1} z^{-1}$ (db)
Kauffman polynomial	$-a z^9-z^9 a^{-1} -a^2 z^8-z^8 a^{-2} -2 z^8-a^5 z^7+8 a z^7+7 z^7 a^{-1} -a^6 z^6-a^4 z^6+8 a^2 z^6+7 z^6 a^{-2} +15 z^6-a^7 z^5+4 a^5 z^5-20 a z^5-15 z^5 a^{-1} +3 a^6 z^4+3 a^4 z^4-18 a^2 z^4-15 z^4 a^{-2} -33 z^4+4 a^7 z^3-5 a^5 z^3+22 a z^3+13 z^3 a^{-1} -a^6 z^2-2 a^4 z^2+13 a^2 z^2+11 z^2 a^{-2} +25 z^2-3 a^7 z+4 a^5 z-13 a z-6 z a^{-1} +a^4-3 a^2-2 a^{-2} -5-a^5 z^{-1} +2 a z^{-1} + a^{-1} z^{-1}$ (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 3 4 5 χ 8                       1 -1 6                         0 4                   1 1   0 2               1 1       0 0             1   1       2 -2           2 3 1         0 -4         1               1 -6       1 2 1             0 -8     1 1                 0 -10     1                   1 -12 1 1                     0 -14 1                       1

Integral Khovanov Homology (db, data source)

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

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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