L11n351

L11n350

L11n352

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Link Presentations

[edit Notes on L11n351's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{-u v^3 w+u v^3+u v^2 w^3-3 u v^2 w^2+4 u v^2 w-2 u v^2-u v w^3+4 u v w^2-3 u v w+u v-v^2 w^3+3 v^2 w^2-4 v^2 w+v^2+2 v w^3-4 v w^2+3 v w-v-w^3+w^2}{\sqrt{u} v^{3/2} w^{3/2}}$ (db)
Jones polynomial	$-q^5+3 q^4-7 q^3+12 q^2-13 q+15-13 q^{-1} +11 q^{-2} -6 q^{-3} +3 q^{-4}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^2 a^{-4} +a^4 z^{-2} +2 a^4- a^{-4} +a^2 z^4+2 z^4 a^{-2} -2 a^2 z^2+3 z^2 a^{-2} -2 a^2 z^{-2} -5 a^2+2 a^{-2} -z^6-2 z^4-z^2+ z^{-2} +2$ (db)
Kauffman polynomial	$2 a z^9+2 z^9 a^{-1} +5 a^2 z^8+5 z^8 a^{-2} +10 z^8+3 a^3 z^7+5 a z^7+7 z^7 a^{-1} +5 z^7 a^{-3} -12 a^2 z^6-5 z^6 a^{-2} +3 z^6 a^{-4} -20 z^6-3 a^3 z^5-13 a z^5-18 z^5 a^{-1} -7 z^5 a^{-3} +z^5 a^{-5} +6 a^4 z^4+25 a^2 z^4-5 z^4 a^{-4} +24 z^4+4 a^3 z^3+12 a z^3+12 z^3 a^{-1} +2 z^3 a^{-3} -2 z^3 a^{-5} -11 a^4 z^2-29 a^2 z^2-z^2 a^{-2} +3 z^2 a^{-4} -22 z^2-5 a^3 z-8 a z-3 z a^{-1} +z a^{-3} +z a^{-5} +6 a^4+13 a^2- a^{-4} +9+2 a^3 z^{-1} +2 a z^{-1} -a^4 z^{-2} -2 a^2 z^{-2} - z^{-2}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 11                   1 -1 9                 2   2 7               5 1   -4 5             7 2     5 3           6 5       -1 1         9 7         2 -1       7 9           2 -3     4 6             -2 -5   2 7               5 -7 1 4                 -3 -9 3                   3

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-4$ ${\mathbb Z}^{3}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=0$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{9}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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