L11n111

L11n110

L11n112

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Link Presentations

[edit Notes on L11n111's Link Presentations]

Planar diagram presentation	X6172 X3,12,4,13 X7,18,8,19 X19,22,20,5 X9,21,10,20 X21,9,22,8 X16,11,17,12 X14,17,15,18 X10,15,11,16 X2536 X13,4,14,1
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 6, -5, -9, 7, 2, -11, -8, 9, -7, 8, 3, -4, 5, -6, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{t(1) t(2)^5-2 t(1) t(2)^4-t(2)^4-t(1) t(2)^3+2 t(2)^3+2 t(1) t(2)^2-t(2)^2-t(1) t(2)-2 t(2)+1}{\sqrt{t(1)} t(2)^{5/2}}$ (db)
Jones polynomial	$-\frac{1}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{2}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{1}{q^{11/2}}+\frac{1}{q^{13/2}}-\frac{2}{q^{15/2}}+\frac{2}{q^{17/2}}-\frac{2}{q^{19/2}}+\frac{1}{q^{21/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{11} z^{-1} +4 z a^9+4 a^9 z^{-1} -z^5 a^7-7 z^3 a^7-11 z a^7-6 a^7 z^{-1} +z^7 a^5+6 z^5 a^5+11 z^3 a^5+11 z a^5+5 a^5 z^{-1} -z^5 a^3-5 z^3 a^3-6 z a^3-2 a^3 z^{-1}$ (db)
Kauffman polynomial	$a^{12} z^6-4 a^{12} z^4+3 a^{12} z^2-a^{12}+2 a^{11} z^7-9 a^{11} z^5+9 a^{11} z^3-4 a^{11} z+a^{11} z^{-1} +a^{10} z^8-3 a^{10} z^6-4 a^{10} z^4+9 a^{10} z^2-3 a^{10}+3 a^9 z^7-18 a^9 z^5+29 a^9 z^3-18 a^9 z+4 a^9 z^{-1} +a^8 z^8-5 a^8 z^6+a^8 z^4+7 a^8 z^2-3 a^8+3 a^7 z^7-22 a^7 z^5+42 a^7 z^3-28 a^7 z+6 a^7 z^{-1} +a^6 z^8-6 a^6 z^6+6 a^6 z^4+a^6 z^2-a^6+3 a^5 z^7-19 a^5 z^5+33 a^5 z^3-22 a^5 z+5 a^5 z^{-1} +a^4 z^8-5 a^4 z^6+5 a^4 z^4-a^4+a^3 z^7-6 a^3 z^5+11 a^3 z^3-8 a^3 z+2 a^3 z^{-1}$ (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 1 2 χ 0                       1 1 -2                         0 -4                   2 1   1 -6               1 1 1     1 -8             1 1 1       -1 -10           2 2 1         1 -12         2 3 1           0 -14       1 1 1             1 -16     1 2 1               0 -18   1 1                   0 -20   1                     1 -22 1                       -1

Integral Khovanov Homology (db, data source)

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

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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