L11n165

L11n164

L11n166

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Link Presentations

[edit Notes on L11n165's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X15,20,16,21 X14,5,15,6 X4,13,5,14 X17,22,18,7 X21,16,22,17 X19,12,20,13 X11,18,12,19 X2738 X6,9,1,10
Gauss code	{1, -10, 2, -5, 4, -11}, {10, -1, 11, -2, -9, 8, 5, -4, -3, 7, -6, 9, -8, 3, -7, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{u^2 v^5-u^2 v^3+u^2 v^2+u v^6-u v^5+u v^3-u v+u+v^4-v^3+v}{u v^3}$ (db)
Jones polynomial	$\frac{1}{q^{9/2}}-\frac{1}{q^{7/2}}+\frac{1}{q^{27/2}}-\frac{1}{q^{25/2}}+\frac{1}{q^{23/2}}-\frac{1}{q^{21/2}}-\frac{1}{q^{15/2}}+\frac{1}{q^{13/2}}-\frac{2}{q^{11/2}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$-2 z a^{13}-2 a^{13} z^{-1} +z^5 a^{11}+7 z^3 a^{11}+11 z a^{11}+5 a^{11} z^{-1} -z^7 a^9-6 z^5 a^9-10 z^3 a^9-7 z a^9-3 a^9 z^{-1} -z^7 a^7-6 z^5 a^7-10 z^3 a^7-5 z a^7$ (db)
Kauffman polynomial	$a^{16} z^6-5 a^{16} z^4+6 a^{16} z^2-a^{16}+a^{15} z^7-5 a^{15} z^5+6 a^{15} z^3-2 a^{15} z+a^{14} z^6-5 a^{14} z^4+4 a^{14} z^2-3 a^{13} z^3+4 a^{13} z-2 a^{13} z^{-1} +a^{12} z^8-8 a^{12} z^6+19 a^{12} z^4-19 a^{12} z^2+5 a^{12}+a^{11} z^9-8 a^{11} z^7+22 a^{11} z^5-31 a^{11} z^3+21 a^{11} z-5 a^{11} z^{-1} +2 a^{10} z^8-13 a^{10} z^6+24 a^{10} z^4-17 a^{10} z^2+5 a^{10}+a^9 z^9-6 a^9 z^7+11 a^9 z^5-12 a^9 z^3+10 a^9 z-3 a^9 z^{-1} +a^8 z^8-5 a^8 z^6+5 a^8 z^4+a^7 z^7-6 a^7 z^5+10 a^7 z^3-5 a^7 z$ (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -6                       1 1 -8                     1 1 0 -10                   1     1 -12               1 1 1     1 -14             1 2 1       0 -16           1 1 1         1 -18         1 3 2           0 -20       1                 1 -22       1 1               0 -24   1 1                   0 -26                         0 -28 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-8$ $i=-6$ $i=-4$ $r=-11$ ${\mathbb Z}$ $r=-10$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}_2$ ${\mathbb Z}^{3}$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

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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