L11n395

L11n394

L11n396

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Link Presentations

[edit Notes on L11n395's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X15,1,16,4 X9,22,10,19 X3849 X21,17,22,16 X11,5,12,18 X5,21,6,20 X17,11,18,10 X19,12,20,13 X2,14,3,13
Gauss code	{1, -11, -5, 3}, {-10, 8, -6, 4}, {-8, -1, 2, 5, -4, 9, -7, 10, 11, -2, -3, 6, -9, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(u-1) (v-1) (w-1) \left(w^2-w+1\right)}{\sqrt{u} \sqrt{v} w^{3/2}}$ (db)
Jones polynomial	$q^5-4 q^4- q^{-4} +6 q^3+3 q^{-3} -6 q^2-4 q^{-2} +9 q+7 q^{-1} -7$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^6-a^2 z^4-2 z^4 a^{-2} +4 z^4-2 a^2 z^2-4 z^2 a^{-2} +z^2 a^{-4} +5 z^2+a^2 z^{-2} + a^{-2} z^{-2} -2 z^{-2}$ (db)
Kauffman polynomial	$2 a z^9+2 z^9 a^{-1} +3 a^2 z^8+5 z^8 a^{-2} +8 z^8+a^3 z^7-5 a z^7-z^7 a^{-1} +5 z^7 a^{-3} -14 a^2 z^6-18 z^6 a^{-2} +2 z^6 a^{-4} -34 z^6-4 a^3 z^5-5 a z^5-17 z^5 a^{-1} -16 z^5 a^{-3} +19 a^2 z^4+22 z^4 a^{-2} -z^4 a^{-4} +42 z^4+4 a^3 z^3+13 a z^3+22 z^3 a^{-1} +17 z^3 a^{-3} +4 z^3 a^{-5} -8 a^2 z^2-13 z^2 a^{-2} -z^2 a^{-4} +z^2 a^{-6} -19 z^2-a^3 z-4 a z-6 z a^{-1} -4 z a^{-3} -z a^{-5} +1-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 11                   1 1 9                 3   -3 7               4 2   2 5             3 3     0 3           6 4 1     3 1         5 7         2 -1       2 3 1         0 -3     2 5             3 -5   1 2               -1 -7   2                 2 -9 1                   -1

Integral Khovanov Homology (db, data source)

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

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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