L11n106

L11n105

L11n107

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Link Presentations

[edit Notes on L11n106's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X7,16,8,17 X17,22,18,5 X13,18,14,19 X9,21,10,20 X19,14,20,15 X21,9,22,8 X15,10,16,11 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 8, -6, 9, 11, -2, -5, 7, -9, 3, -4, 5, -7, 6, -8, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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