L11n274

L11n273

L11n275

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Link Presentations

[edit Notes on L11n274's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X13,2,14,3 X14,7,15,8 X9,18,10,19 X17,11,18,22 X11,21,12,20 X21,17,22,16 X4,15,1,16 X19,10,20,5
Gauss code	{1, 4, -3, -10}, {-2, -1, 5, 3, -6, 11}, {-8, 2, -4, -5, 10, 9, -7, 6, -11, 8, -9, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{u v^2 w^4-2 u v^2 w^3+2 u v^2 w^2+u v w^3-u v w^2+v w^2-v w-2 w^2+2 w-1}{\sqrt{u} v w^2}$ (db)
Jones polynomial	$-q+2-3 q^{-1} +4 q^{-2} -4 q^{-3} +5 q^{-4} -3 q^{-5} +4 q^{-6} - q^{-7} + q^{-8}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^8 z^2+2 a^8 z^{-2} +2 a^8-a^6 z^6-6 a^6 z^4-12 a^6 z^2-5 a^6 z^{-2} -12 a^6+a^4 z^8+7 a^4 z^6+18 a^4 z^4+23 a^4 z^2+4 a^4 z^{-2} +14 a^4-a^2 z^6-5 a^2 z^4-7 a^2 z^2-a^2 z^{-2} -4 a^2$ (db)
Kauffman polynomial	$a^{10} z^2-a^{10}+a^9 z^3+3 a^8 z^4-8 a^8 z^2-2 a^8 z^{-2} +8 a^8+2 a^7 z^7-10 a^7 z^5+20 a^7 z^3-18 a^7 z+5 a^7 z^{-1} +3 a^6 z^8-16 a^6 z^6+31 a^6 z^4-33 a^6 z^2-5 a^6 z^{-2} +20 a^6+a^5 z^9-19 a^5 z^5+41 a^5 z^3-33 a^5 z+9 a^5 z^{-1} +5 a^4 z^8-26 a^4 z^6+42 a^4 z^4-32 a^4 z^2-4 a^4 z^{-2} +15 a^4+a^3 z^9-a^3 z^7-14 a^3 z^5+29 a^3 z^3-19 a^3 z+5 a^3 z^{-1} +2 a^2 z^8-10 a^2 z^6+14 a^2 z^4-8 a^2 z^2-a^2 z^{-2} +3 a^2+a z^7-5 a z^5+7 a z^3-4 a z+a z^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 3                   1 -1 1                 1   1 -1               2 1   -1 -3           1 3 1     1 -5           3 3       0 -7       1 4 2         1 -9     1 2 3           2 -11     4 3             1 -13 1   2               3 -15 2 2                 0 -17 1                   1

Integral Khovanov Homology (db, data source)

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

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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