L11n256

L11n255

L11n257

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Link Presentations

[edit Notes on L11n256's Link Presentations]

Planar diagram presentation	X6172 X3,11,4,10 X16,12,17,11 X18,22,19,21 X20,14,21,13 X12,20,13,19 X22,18,9,17 X15,8,16,5 X7,14,8,15 X2536 X9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -9, 8}, {-11, 2, 3, -6, 5, 9, -8, -3, 7, -4, 6, -5, 4, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{2 t(1) t(3)^3-2 t(1) t(3)^2-t(1) t(2) t(3)^2-t(2) t(3)^2-t(3)^2+t(1) t(3)+t(1) t(2) t(3)+2 t(2) t(3)+t(3)-2 t(2)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}}$ (db)
Jones polynomial	$-q^7+q^6-q^5-q^4+2 q^3+ q^{-3} -q^2- q^{-2} +4 q+3 q^{-1} -2$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^4 a^{-2} +z^4 a^{-4} -z^4+a^2 z^2-3 z^2 a^{-2} +5 z^2 a^{-4} -z^2 a^{-6} -3 z^2+2 a^2-3 a^{-2} +6 a^{-4} -2 a^{-6} -3+a^2 z^{-2} -2 a^{-2} z^{-2} +3 a^{-4} z^{-2} - a^{-6} z^{-2} - z^{-2}$ (db)
Kauffman polynomial	$z^7 a^{-7} -6 z^5 a^{-7} +10 z^3 a^{-7} -7 z a^{-7} +2 a^{-7} z^{-1} +z^8 a^{-6} -6 z^6 a^{-6} +8 z^4 a^{-6} -2 z^2 a^{-6} - a^{-6} z^{-2} + a^{-6} +2 z^7 a^{-5} -16 z^5 a^{-5} +35 z^3 a^{-5} -27 z a^{-5} +8 a^{-5} z^{-1} +z^8 a^{-4} -7 z^6 a^{-4} +12 z^4 a^{-4} -7 z^2 a^{-4} -3 a^{-4} z^{-2} +5 a^{-4} +3 z^7 a^{-3} -21 z^5 a^{-3} +44 z^3 a^{-3} -34 z a^{-3} +10 a^{-3} z^{-1} +z^8 a^{-2} +a^2 z^6-5 z^6 a^{-2} -5 a^2 z^4+8 z^4 a^{-2} +7 a^2 z^2-8 z^2 a^{-2} +a^2 z^{-2} -2 a^{-2} z^{-2} -4 a^2+4 a^{-2} +a z^7+3 z^7 a^{-1} -3 a z^5-14 z^5 a^{-1} -a z^3+18 z^3 a^{-1} +4 a z-10 z a^{-1} -2 a z^{-1} +2 a^{-1} z^{-1} +z^8-3 z^6-z^4+4 z^2+ z^{-2} -3$ (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 7 χ 15                       1 -1 13                         0 11                   1 1   0 9               3 1       -2 7             1 1 1       1 5           2 4 1         1 3         3 1 1           3 1       2 5 1             2 -1     1 1 1               1 -3     2                   2 -5 1 1                     0 -7 1                       1

Integral Khovanov Homology (db, data source)

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

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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