L11n366

L11n365

L11n367

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Link Presentations

[edit Notes on L11n366's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,8,13,7 X15,17,16,22 X18,10,19,9 X8,18,9,17 X13,21,14,20 X21,15,22,14 X19,5,20,16 X2536 X4,12,1,11
Gauss code	{1, -10, 2, -11}, {6, -5, -9, 7, -8, 4}, {10, -1, 3, -6, 5, -2, 11, -3, -7, 8, -4, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{(u-1) (w-1) \left(v^2 w^3-1\right)}{\sqrt{u} v w^2}$ (db)
Jones polynomial	$q^9-q^8+2 q^7-2 q^6+3 q^5-2 q^4+3 q^3-q^2+q$ (db)
Signature	6 (db)
HOMFLY-PT polynomial	$-z^8 a^{-6} +z^6 a^{-4} -7 z^6 a^{-6} +z^6 a^{-8} +6 z^4 a^{-4} -17 z^4 a^{-6} +6 z^4 a^{-8} +11 z^2 a^{-4} -20 z^2 a^{-6} +10 z^2 a^{-8} -z^2 a^{-10} +7 a^{-4} -12 a^{-6} +6 a^{-8} - a^{-10} + a^{-4} z^{-2} -2 a^{-6} z^{-2} + a^{-8} z^{-2}$ (db)
Kauffman polynomial	$a^{-12} -z a^{-11} +z^4 a^{-10} -5 z^2 a^{-10} +2 a^{-10} +2 z^7 a^{-9} -10 z^5 a^{-9} +11 z^3 a^{-9} -3 z a^{-9} +3 z^8 a^{-8} -18 z^6 a^{-8} +33 z^4 a^{-8} -26 z^2 a^{-8} - a^{-8} z^{-2} +9 a^{-8} +z^9 a^{-7} -3 z^7 a^{-7} -6 z^5 a^{-7} +19 z^3 a^{-7} -12 z a^{-7} +2 a^{-7} z^{-1} +4 z^8 a^{-6} -25 z^6 a^{-6} +49 z^4 a^{-6} -39 z^2 a^{-6} -2 a^{-6} z^{-2} +15 a^{-6} +z^9 a^{-5} -5 z^7 a^{-5} +4 z^5 a^{-5} +8 z^3 a^{-5} -10 z a^{-5} +2 a^{-5} z^{-1} +z^8 a^{-4} -7 z^6 a^{-4} +17 z^4 a^{-4} -18 z^2 a^{-4} - a^{-4} z^{-2} +8 a^{-4}$ (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   \ -2 -1 0 1 2 3 4 5 6 χ 19                 1 1 17               2 2 0 15             1 1 1 1 13           2 2     0 11         1 1 1     1 9       1 2         1 7     2 1           1 5   1 3             2 3                   0 1 1                 1

Integral Khovanov Homology (db, data source)

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

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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