L11n30

L11n29

L11n31

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Link Presentations

[edit Notes on L11n30's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X19,1,20,4 X5,12,6,13 X3849 X9,16,10,17 X13,22,14,5 X15,10,16,11 X21,14,22,15 X11,20,12,21 X2,18,3,17
Gauss code	{1, -11, -5, 3}, {-4, -1, 2, 5, -6, 8, -10, 4, -7, 9, -8, 6, 11, -2, -3, 10, -9, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{2 (u-1) (v-1)}{\sqrt{u} \sqrt{v}}$ (db)
Jones polynomial	$-\frac{3}{q^{9/2}}+\frac{1}{q^{7/2}}-\frac{2}{q^{5/2}}+\frac{1}{q^{3/2}}+\frac{1}{q^{19/2}}-\frac{1}{q^{17/2}}+\frac{2}{q^{15/2}}-\frac{2}{q^{13/2}}+\frac{2}{q^{11/2}}-\frac{1}{\sqrt{q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-z a^9-a^9 z^{-1} +z^3 a^7+2 z a^7+a^7 z^{-1} +z^3 a^5+2 z a^5+2 a^5 z^{-1} -2 z a^3-2 a^3 z^{-1} -z a$ (db)
Kauffman polynomial	$a^{10} z^8-7 a^{10} z^6+16 a^{10} z^4-14 a^{10} z^2+4 a^{10}+a^9 z^9-6 a^9 z^7+10 a^9 z^5-4 a^9 z^3-a^9 z^{-1} +3 a^8 z^8-19 a^8 z^6+38 a^8 z^4-30 a^8 z^2+9 a^8+a^7 z^9-4 a^7 z^7+8 a^7 z^3-3 a^7 z-a^7 z^{-1} +2 a^6 z^8-11 a^6 z^6+18 a^6 z^4-13 a^6 z^2+4 a^6+2 a^5 z^7-10 a^5 z^5+14 a^5 z^3-9 a^5 z+2 a^5 z^{-1} +a^4 z^6-4 a^4 z^4+4 a^4 z^2-2 a^4+2 a^3 z^3-5 a^3 z+2 a^3 z^{-1} +a^2 z^2+a z$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ 0                   1 1 -2                 2 2 0 -4               1 1 1 1 -6             2 3     1 -8           2 1 1     2 -10         1 3 1       1 -12       2 2           0 -14       1 1           0 -16   1 2               -1 -18                     0 -20 1                   -1

Integral Khovanov Homology (db, data source)

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

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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