L11n220

L11n219

L11n221

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Link Presentations

[edit Notes on L11n220's Link Presentations]

Planar diagram presentation	X10,1,11,2 X8,9,1,10 X3,12,4,13 X15,22,16,9 X2,17,3,18 X21,4,22,5 X14,5,15,6 X20,13,21,14 X11,16,12,17 X6,19,7,20 X18,7,19,8
Gauss code	{1, -5, -3, 6, 7, -10, 11, -2}, {2, -1, -9, 3, 8, -7, -4, 9, 5, -11, 10, -8, -6, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{t(1)^2 t(2)^5+t(1)^3 t(2)^4-t(1)^2 t(2)^4+t(1)^2 t(2)^3+t(1) t(2)^2-t(1) t(2)+t(2)+t(1)}{t(1)^{3/2} t(2)^{5/2}}$ (db)
Jones polynomial	$-\frac{1}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{2}{q^{11/2}}+\frac{2}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{2}{q^{17/2}}-\frac{2}{q^{19/2}}+\frac{2}{q^{21/2}}-\frac{1}{q^{23/2}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$a^{13} (-z)-a^{13} z^{-1} +a^{11} z^5+6 a^{11} z^3+9 a^{11} z+3 a^{11} z^{-1} -a^9 z^7-6 a^9 z^5-11 a^9 z^3-8 a^9 z-2 a^9 z^{-1} -a^7 z^7-6 a^7 z^5-10 a^7 z^3-4 a^7 z$ (db)
Kauffman polynomial	$a^{15} z+a^{14} z^2+a^{14}+a^{13} z^5-3 a^{13} z^3+3 a^{13} z-a^{13} z^{-1} +a^{12} z^8-6 a^{12} z^6+13 a^{12} z^4-12 a^{12} z^2+3 a^{12}+a^{11} z^9-7 a^{11} z^7+20 a^{11} z^5-29 a^{11} z^3+16 a^{11} z-3 a^{11} z^{-1} +2 a^{10} z^8-11 a^{10} z^6+19 a^{10} z^4-14 a^{10} z^2+3 a^{10}+a^9 z^9-6 a^9 z^7+13 a^9 z^5-16 a^9 z^3+10 a^9 z-2 a^9 z^{-1} +a^8 z^8-5 a^8 z^6+6 a^8 z^4-a^8 z^2+a^7 z^7-6 a^7 z^5+10 a^7 z^3-4 a^7 z$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ -6                 1 1 -8               1 1 0 -10             1     1 -12           1 1     0 -14         2 1       1 -16         1         1 -18     2 2           0 -20                   0 -22 1 2               -1 -24 1                 1

Integral Khovanov Homology (db, data source)

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

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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