L11a411

L11a410

L11a412

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Link Presentations

[edit Notes on L11a411's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X22,16,11,15 X14,8,15,7 X10,12,5,11 X20,18,21,17 X8,19,9,20 X18,9,19,10 X16,22,17,21 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -7, 8, -5}, {5, -2, 11, -4, 3, -9, 6, -8, 7, -6, 9, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{2 u v^2 w^2-2 u v^2 w-4 u v w^2+5 u v w-2 u v+2 u w^2-4 u w+2 u-2 v^2 w^2+4 v^2 w-2 v^2+2 v w^2-5 v w+4 v+2 w-2}{\sqrt{u} v w}$ (db)
Jones polynomial	$-q^8+3 q^7-6 q^6+10 q^5-13 q^4+15 q^3-14 q^2+13 q-8+6 q^{-1} -2 q^{-2} + q^{-3}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^6 a^{-2} +z^6 a^{-4} +2 z^4 a^{-2} +3 z^4 a^{-4} -z^4 a^{-6} -2 z^4+a^2 z^2+z^2 a^{-2} +4 z^2 a^{-4} -2 z^2 a^{-6} -5 z^2+2 a^2+ a^{-2} +2 a^{-4} - a^{-6} -4+a^2 z^{-2} + a^{-2} z^{-2} -2 z^{-2}$ (db)
Kauffman polynomial	$z^{10} a^{-2} +z^{10} a^{-4} +2 z^9 a^{-1} +6 z^9 a^{-3} +4 z^9 a^{-5} +3 z^8 a^{-2} +7 z^8 a^{-4} +6 z^8 a^{-6} +2 z^8+2 a z^7-z^7 a^{-1} -15 z^7 a^{-3} -7 z^7 a^{-5} +5 z^7 a^{-7} +a^2 z^6-9 z^6 a^{-2} -27 z^6 a^{-4} -16 z^6 a^{-6} +3 z^6 a^{-8} -5 a z^5-2 z^5 a^{-1} +18 z^5 a^{-3} +3 z^5 a^{-5} -11 z^5 a^{-7} +z^5 a^{-9} -4 a^2 z^4+7 z^4 a^{-2} +41 z^4 a^{-4} +19 z^4 a^{-6} -6 z^4 a^{-8} -13 z^4+a z^3-6 z^3 a^{-1} -8 z^3 a^{-3} +8 z^3 a^{-5} +7 z^3 a^{-7} -2 z^3 a^{-9} +6 a^2 z^2-3 z^2 a^{-2} -22 z^2 a^{-4} -9 z^2 a^{-6} +z^2 a^{-8} +15 z^2+4 a z+6 z a^{-1} -4 z a^{-5} -2 z a^{-7} -4 a^2-2 a^{-2} +4 a^{-4} +2 a^{-6} -7-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2}$ (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 6 7 χ 17                       1 -1 15                     2   2 13                   4 1   -3 11                 6 2     4 9               7 4       -3 7             8 6         2 5           7 8           1 3         6 7             -1 1       4 9               5 -1     2 4                 -2 -3   1 5                   4 -5   1                     -1 -7 1                       1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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