L11n76

L11n75

L11n77

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Link Presentations

[edit Notes on L11n76's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{t(1) t(2)^5-5 t(1) t(2)^4+t(2)^4+8 t(1) t(2)^3-6 t(2)^3-6 t(1) t(2)^2+8 t(2)^2+t(1) t(2)-5 t(2)+1}{\sqrt{t(1)} t(2)^{5/2}}$ (db)
Jones polynomial	$q^{9/2}-\frac{3}{q^{9/2}}-3 q^{7/2}+\frac{7}{q^{7/2}}+7 q^{5/2}-\frac{11}{q^{5/2}}-11 q^{3/2}+\frac{13}{q^{3/2}}+13 \sqrt{q}-\frac{15}{\sqrt{q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^5 z+a^5 z^{-1} -a^3 z^5-3 a^3 z^3+z^3 a^{-3} -4 a^3 z+2 z a^{-3} -2 a^3 z^{-1} + a^{-3} z^{-1} +a z^7+4 a z^5-2 z^5 a^{-1} +6 a z^3-6 z^3 a^{-1} +5 a z-6 z a^{-1} +3 a z^{-1} -3 a^{-1} z^{-1}$ (db)
Kauffman polynomial	$-2 a z^9-2 z^9 a^{-1} -7 a^2 z^8-4 z^8 a^{-2} -11 z^8-8 a^3 z^7-11 a z^7-6 z^7 a^{-1} -3 z^7 a^{-3} -3 a^4 z^6+12 a^2 z^6+7 z^6 a^{-2} -z^6 a^{-4} +23 z^6+16 a^3 z^5+35 a z^5+27 z^5 a^{-1} +8 z^5 a^{-3} -3 a^4 z^4-13 a^2 z^4+2 z^4 a^{-2} +3 z^4 a^{-4} -11 z^4-6 a^5 z^3-21 a^3 z^3-33 a z^3-25 z^3 a^{-1} -7 z^3 a^{-3} +3 a^4 z^2+9 a^2 z^2-6 z^2 a^{-2} -3 z^2 a^{-4} +3 z^2+5 a^5 z+11 a^3 z+15 a z+12 z a^{-1} +3 z a^{-3} -2 a^2+2 a^{-2} + a^{-4} -a^5 z^{-1} -2 a^3 z^{-1} -3 a z^{-1} -3 a^{-1} z^{-1} - a^{-3} z^{-1}$ (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 χ 10                   1 -1 8                 2   2 6               5 1   -4 4             6 2     4 2           7 5       -2 0         8 6         2 -2       6 8           2 -4     5 7             -2 -6   2 6               4 -8 1 5                 -4 -10 3                   3

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-4$ ${\mathbb Z}^{3}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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