L11n172

L11n171

L11n173

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Link Presentations

[edit Notes on L11n172's Link Presentations]

Planar diagram presentation	X8192 X3,10,4,11 X12,7,13,8 X22,15,7,16 X14,6,15,5 X6,14,1,13 X16,21,17,22 X9,18,10,19 X20,11,21,12 X4,18,5,17 X19,3,20,2
Gauss code	{1, 11, -2, -10, 5, -6}, {3, -1, -8, 2, 9, -3, 6, -5, 4, -7, 10, 8, -11, -9, 7, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{u^2 v^2-3 u^2 v+2 u^2-5 u v^2+9 u v-5 u+2 v^2-3 v+1}{u v}$ (db)
Jones polynomial	$-\frac{8}{q^{9/2}}+\frac{10}{q^{7/2}}-\frac{11}{q^{5/2}}-2 q^{3/2}+\frac{9}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{3}{q^{13/2}}+\frac{6}{q^{11/2}}+4 \sqrt{q}-\frac{8}{\sqrt{q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-z a^7+2 z^3 a^5+z a^5-a^5 z^{-1} -z^5 a^3-z^3 a^3+2 a^3 z^{-1} +3 z^3 a+3 z a-2 z a^{-1} - a^{-1} z^{-1}$ (db)
Kauffman polynomial	$a^8 z^6-3 a^8 z^4+2 a^8 z^2+3 a^7 z^7-9 a^7 z^5+6 a^7 z^3-a^7 z+4 a^6 z^8-12 a^6 z^6+10 a^6 z^4-6 a^6 z^2+2 a^6+2 a^5 z^9-a^5 z^7-8 a^5 z^5+5 a^5 z^3-a^5 z^{-1} +8 a^4 z^8-25 a^4 z^6+29 a^4 z^4-18 a^4 z^2+5 a^4+2 a^3 z^9-a^3 z^7-4 a^3 z^5+4 a^3 z^3+2 a^3 z-2 a^3 z^{-1} +4 a^2 z^8-11 a^2 z^6+18 a^2 z^4-11 a^2 z^2+3 a^2+3 a z^7-5 a z^5+8 a z^3+3 z^3 a^{-1} -3 a z-4 z a^{-1} + a^{-1} z^{-1} +z^6+2 z^4-z^2-1$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 4                   2 2 2                 3 1 -2 0               5 1   4 -2             5 4     -1 -4           6 4       2 -6         4 5         1 -8       4 6           -2 -10     2 4             2 -12   1 4               -3 -14   2                 2 -16 1                   -1

Integral Khovanov Homology (db, data source)

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

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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