L11n167

L11n166

L11n168

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Link Presentations

[edit Notes on L11n167's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X11,19,12,18 X22,19,7,20 X20,15,21,16 X16,21,17,22 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 $u$, $v$, $w$, ...)	$\frac{t(1)^2 t(2)^4-t(1) t(2)^4-3 t(1)^2 t(2)^3+3 t(1) t(2)^3-t(2)^3+2 t(1)^2 t(2)^2-5 t(1) t(2)^2+2 t(2)^2-t(1)^2 t(2)+3 t(1) t(2)-3 t(2)-t(1)+1}{t(1) t(2)^2}$ (db)
Jones polynomial	$-\frac{2}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{8}{q^{11/2}}-\frac{9}{q^{13/2}}+\frac{9}{q^{15/2}}-\frac{8}{q^{17/2}}+\frac{5}{q^{19/2}}-\frac{3}{q^{21/2}}+\frac{1}{q^{23/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^9 \left(-z^5\right)-3 a^9 z^3-2 a^9 z+a^7 z^7+5 a^7 z^5+9 a^7 z^3+7 a^7 z+a^7 z^{-1} -2 a^5 z^5-8 a^5 z^3-8 a^5 z-a^5 z^{-1}$ (db)
Kauffman polynomial	$-z^4 a^{14}+z^2 a^{14}-3 z^5 a^{13}+4 z^3 a^{13}-z a^{13}-4 z^6 a^{12}+4 z^4 a^{12}-4 z^7 a^{11}+4 z^5 a^{11}-2 z a^{11}-3 z^8 a^{10}+4 z^6 a^{10}-4 z^4 a^{10}+z^2 a^{10}-z^9 a^9-2 z^7 a^9+5 z^5 a^9-3 z^3 a^9-4 z^8 a^8+10 z^6 a^8-11 z^4 a^8+6 z^2 a^8-z^9 a^7+2 z^7 a^7-5 z^5 a^7+11 z^3 a^7-7 z a^7+a^7 z^{-1} -z^8 a^6+2 z^6 a^6-2 z^4 a^6+4 z^2 a^6-a^6-3 z^5 a^5+10 z^3 a^5-8 z a^5+a^5 z^{-1}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                   2 2 -6                 2 1 -1 -8               4 1   3 -10             4 2     -2 -12           5 4       1 -14         4 4         0 -16       4 5           -1 -18     2 5             3 -20   1 3               -2 -22   2                 2 -24 1                   -1

Integral Khovanov Homology (db, data source)

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

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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