L11n404

L11n403

L11n405

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Link Presentations

[edit Notes on L11n404's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X16,9,17,10 X8,15,9,16 X4,17,1,18 X11,22,12,19 X10,4,11,3 X5,21,6,20 X21,5,22,18 X19,12,20,13 X2,14,3,13
Gauss code	{1, -11, 7, -5}, {-10, 8, -9, 6}, {-8, -1, 2, -4, 3, -7, -6, 10, 11, -2, 4, -3, 5, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$0$ (db)
Jones polynomial	$q^2-q+2+2 q^{-2} + q^{-3} - q^{-4} + q^{-5} - q^{-6} + q^{-7} - q^{-8}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^6 \left(-z^4\right)-4 a^6 z^2-a^6 z^{-2} -4 a^6+a^4 z^6+7 a^4 z^4+16 a^4 z^2+4 a^4 z^{-2} +13 a^4-a^2 z^6-7 a^2 z^4-16 a^2 z^2-5 a^2 z^{-2} -14 a^2+z^4+4 z^2+2 z^{-2} +5$ (db)
Kauffman polynomial	$z^5 a^9-4 z^3 a^9+2 z a^9+z^6 a^8-4 z^4 a^8+2 z^2 a^8+z^7 a^7-5 z^5 a^7+6 z^3 a^7-4 z a^7+a^7 z^{-1} +z^8 a^6-6 z^6 a^6+10 z^4 a^6-10 z^2 a^6-a^6 z^{-2} +7 a^6+3 z^7 a^5-21 z^5 a^5+40 z^3 a^5-25 z a^5+5 a^5 z^{-1} +3 z^8 a^4-22 z^6 a^4+50 z^4 a^4-49 z^2 a^4-4 a^4 z^{-2} +22 a^4+z^9 a^3-4 z^7 a^3-7 z^5 a^3+34 z^3 a^3-31 z a^3+9 a^3 z^{-1} +3 z^8 a^2-22 z^6 a^2+52 z^4 a^2-53 z^2 a^2-5 a^2 z^{-2} +23 a^2+z^9 a-6 z^7 a+8 z^5 a+4 z^3 a-12 z a+5 a z^{-1} +z^8-7 z^6+16 z^4-16 z^2-2 z^{-2} +9$ (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 3 4 χ 5                       1 1 3                         0 1                   2 1   1 -1               3 1       2 -3             1 5 2       2 -5           3 2 2         3 -7         1 2 1           0 -9       2 4 2             0 -11       1 1               0 -13   1 2 1                 0 -15                         0 -17 1                       -1

Integral Khovanov Homology (db, data source)

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

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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