L11n394

L11n393

L11n395

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Link Presentations

[edit Notes on L11n394's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 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 X13,2,14,3
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 $u$, $v$, $w$, ...)	$-\frac{(v-1) (w-1) \left(w^2-w+1\right) \left(u w^2-1\right)}{\sqrt{u} \sqrt{v} w^{5/2}}$ (db)
Jones polynomial	$- q^{-6} +4 q^{-5} -4 q^{-4} +q^3+8 q^{-3} -3 q^2-8 q^{-2} +5 q+8 q^{-1} -6$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^2 z^8+a^4 z^6-6 a^2 z^6+z^6+5 a^4 z^4-12 a^2 z^4+4 z^4-a^6 z^2+7 a^4 z^2-10 a^2 z^2+4 z^2-a^6+2 a^4-3 a^2+2+a^6 z^{-2} -2 a^4 z^{-2} +a^2 z^{-2}$ (db)
Kauffman polynomial	$a^7 z^3-a^7 z+4 a^6 z^4-6 a^6 z^2+a^6 z^{-2} -a^6+2 a^5 z^7-6 a^5 z^5+9 a^5 z^3-a^5 z-2 a^5 z^{-1} +4 a^4 z^8-19 a^4 z^6+37 a^4 z^4-25 a^4 z^2+2 a^4 z^{-2} +2 a^4+2 a^3 z^9-4 a^3 z^7-6 a^3 z^5+19 a^3 z^3-5 a^3 z-2 a^3 z^{-1} +8 a^2 z^8-35 a^2 z^6+z^6 a^{-2} +51 a^2 z^4-3 z^4 a^{-2} -28 a^2 z^2+z^2 a^{-2} +a^2 z^{-2} +5 a^2+2 a z^9-3 a z^7+3 z^7 a^{-1} -10 a z^5-10 z^5 a^{-1} +17 a z^3+6 z^3 a^{-1} -7 a z-2 z a^{-1} +4 z^8-15 z^6+15 z^4-8 z^2+3$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                   1 1 5                 2   -2 3               3 1   2 1             3 2     -1 -1         1 6 3       2 -3         5 5         0 -5       4 4           0 -7   1 2 5             4 -9   3 3               0 -11   3                 3 -13 1                   -1

Integral Khovanov Homology (db, data source)

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

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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