L10a27

L10a26

L10a28

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Link Presentations

[edit Notes on L10a27's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X14,8,15,7 X16,10,17,9 X8,16,9,15 X20,18,5,17 X18,11,19,12 X10,19,11,20 X2536 X4,14,1,13
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -5, 4, -8, 7, -2, 10, -3, 5, -4, 6, -7, 8, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(t(1)-1) (t(2)-1) \left(t(2)^4-2 t(2)^3+t(2)^2-2 t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}}$ (db)
Jones polynomial	$-q^{15/2}+3 q^{13/2}-5 q^{11/2}+7 q^{9/2}-9 q^{7/2}+9 q^{5/2}-8 q^{3/2}+6 \sqrt{q}-\frac{5}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^5 a^{-5} -3 z^3 a^{-5} -2 z a^{-5} - a^{-5} z^{-1} +z^7 a^{-3} +5 z^5 a^{-3} +9 z^3 a^{-3} +8 z a^{-3} +3 a^{-3} z^{-1} -2 z^5 a^{-1} +a z^3-8 z^3 a^{-1} +3 a z-9 z a^{-1} +2 a z^{-1} -4 a^{-1} z^{-1}$ (db)
Kauffman polynomial	$z^3 a^{-9} +3 z^4 a^{-8} -z^2 a^{-8} +5 z^5 a^{-7} -4 z^3 a^{-7} +z a^{-7} +6 z^6 a^{-6} -7 z^4 a^{-6} +2 z^2 a^{-6} - a^{-6} +6 z^7 a^{-5} -10 z^5 a^{-5} +3 z^3 a^{-5} -2 z a^{-5} + a^{-5} z^{-1} +4 z^8 a^{-4} -5 z^6 a^{-4} -8 z^4 a^{-4} +10 z^2 a^{-4} -3 a^{-4} +z^9 a^{-3} +7 z^7 a^{-3} -32 z^5 a^{-3} +34 z^3 a^{-3} -15 z a^{-3} +3 a^{-3} z^{-1} +6 z^8 a^{-2} -19 z^6 a^{-2} +10 z^4 a^{-2} +7 z^2 a^{-2} -3 a^{-2} +z^9 a^{-1} +a z^7+2 z^7 a^{-1} -5 a z^5-22 z^5 a^{-1} +9 a z^3+35 z^3 a^{-1} -7 a z-19 z a^{-1} +2 a z^{-1} +4 a^{-1} z^{-1} +2 z^8-8 z^6+8 z^4-2$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 16                     1 1 14                   2   -2 12                 3 1   2 10               4 2     -2 8             5 3       2 6           4 4         0 4         4 5           -1 2       4 6             2 0     1 2               -1 -2   1 4                 3 -4   1                   -1 -6 1                     1

Integral Khovanov Homology (db, data source)

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

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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