L11n73

L11n72

L11n74

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Link Presentations

[edit Notes on L11n73's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X16,8,17,7 X17,22,18,5 X11,18,12,19 X21,12,22,13 X13,20,14,21 X19,14,20,15 X8,16,9,15 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, -5, 6, -7, 8, 9, -3, -4, 5, -8, 7, -6, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{2 u v^3-5 u v^2+4 u v-2 u-2 v^3+4 v^2-5 v+2}{\sqrt{u} v^{3/2}}$ (db)
Jones polynomial	$\frac{9}{q^{9/2}}-\frac{9}{q^{7/2}}+\frac{7}{q^{5/2}}-\frac{6}{q^{3/2}}+\frac{1}{q^{17/2}}-\frac{3}{q^{15/2}}+\frac{5}{q^{13/2}}-\frac{8}{q^{11/2}}-\sqrt{q}+\frac{3}{\sqrt{q}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^7 \left(-z^3\right)-a^7 z+a^5 z^5+2 a^5 z^3+2 a^5 z+a^5 z^{-1} +a^3 z^5+a^3 z^3-2 a^3 z-a^3 z^{-1} -a z^3-a z$ (db)
Kauffman polynomial	$a^{10} z^4-2 a^{10} z^2+3 a^9 z^5-5 a^9 z^3+a^9 z+a^8 z^8-2 a^8 z^6+5 a^8 z^4-3 a^8 z^2+a^7 z^9-2 a^7 z^7+5 a^7 z^5-2 a^7 z^3+a^7 z+4 a^6 z^8-10 a^6 z^6+12 a^6 z^4-2 a^6 z^2+a^5 z^9+2 a^5 z^7-7 a^5 z^5+8 a^5 z^3-4 a^5 z+a^5 z^{-1} +3 a^4 z^8-5 a^4 z^6+2 a^4 z^4-a^4+4 a^3 z^7-8 a^3 z^5+3 a^3 z^3-3 a^3 z+a^3 z^{-1} +3 a^2 z^6-6 a^2 z^4+a^2 z^2+a z^5-2 a z^3+a z$ (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 2                   1 1 0                 2   -2 -2               4 1   3 -4             4 3     -1 -6           5 3       2 -8         4 4         0 -10       4 5           -1 -12     2 5             3 -14   1 3               -2 -16   2                 2 -18 1                   -1

Integral Khovanov Homology (db, data source)

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

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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