L11n105

L11n104

L11n106

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Link Presentations

[edit Notes on L11n105's Link Presentations]

Planar diagram presentation	X6172 X3,13,4,12 X7,16,8,17 X17,22,18,5 X13,18,14,19 X9,21,10,20 X19,14,20,15 X21,9,22,8 X15,10,16,11 X2536 X11,1,12,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 8, -6, 9, -11, 2, -5, 7, -9, 3, -4, 5, -7, 6, -8, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{(t(1)-1) (t(2)-1)^3 \left(t(2)^2+1\right)}{\sqrt{t(1)} t(2)^{5/2}}$ (db)
Jones polynomial	$\frac{8}{q^{9/2}}-\frac{10}{q^{7/2}}-q^{5/2}+\frac{10}{q^{5/2}}+3 q^{3/2}-\frac{11}{q^{3/2}}+\frac{2}{q^{13/2}}-\frac{5}{q^{11/2}}-6 \sqrt{q}+\frac{8}{\sqrt{q}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-z a^7-a^7 z^{-1} +z^5 a^5+4 z^3 a^5+6 z a^5+3 a^5 z^{-1} -z^7 a^3-5 z^5 a^3-10 z^3 a^3-10 z a^3-3 a^3 z^{-1} +2 z^5 a+7 z^3 a+7 z a+2 a z^{-1} -z^3 a^{-1} -2 z a^{-1} - a^{-1} z^{-1}$ (db)
Kauffman polynomial	$3 a^8 z^2-a^8+a^7 z^5+5 a^7 z^3-3 a^7 z+a^7 z^{-1} +5 a^6 z^6-7 a^6 z^4+6 a^6 z^2-2 a^6+9 a^5 z^7-26 a^5 z^5+28 a^5 z^3-14 a^5 z+3 a^5 z^{-1} +7 a^4 z^8-18 a^4 z^6+11 a^4 z^4-3 a^4 z^2+2 a^3 z^9+7 a^3 z^7-42 a^3 z^5+48 a^3 z^3-22 a^3 z+3 a^3 z^{-1} +10 a^2 z^8-35 a^2 z^6+32 a^2 z^4-10 a^2 z^2+2 a^2+2 a z^9-a z^7+z^7 a^{-1} -19 a z^5-4 z^5 a^{-1} +31 a z^3+6 z^3 a^{-1} -15 a z-4 z a^{-1} +2 a z^{-1} + a^{-1} z^{-1} +3 z^8-12 z^6+14 z^4-4 z^2$ (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 6                   1 1 4                 2   -2 2               4 1   3 0             4 2     -2 -2           7 4       3 -4         5 6         1 -6       5 5           0 -8     3 5             2 -10   2 5               -3 -12   3                 3 -14 2                   -2

Integral Khovanov Homology (db, data source)

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

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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