L11a88

L11a87

L11a89

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Link Presentations

[edit Notes on L11a88's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{2 u v^3-7 u v^2+9 u v-5 u-5 v^3+9 v^2-7 v+2}{\sqrt{u} v^{3/2}}$ (db)
Jones polynomial	$-9 q^{9/2}+13 q^{7/2}-\frac{1}{q^{7/2}}-15 q^{5/2}+\frac{2}{q^{5/2}}+14 q^{3/2}-\frac{6}{q^{3/2}}+q^{15/2}-3 q^{13/2}+6 q^{11/2}-13 \sqrt{q}+\frac{9}{\sqrt{q}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z a^{-7} -2 z^3 a^{-5} -z a^{-5} + a^{-5} z^{-1} +z^5 a^{-3} +a^3 z-z a^{-3} +a^3 z^{-1} -2 a^{-3} z^{-1} +z^5 a^{-1} -2 a z^3-2 a z-a z^{-1} + a^{-1} z^{-1}$ (db)
Kauffman polynomial	$z^6 a^{-8} -3 z^4 a^{-8} +z^2 a^{-8} +3 z^7 a^{-7} -9 z^5 a^{-7} +5 z^3 a^{-7} -z a^{-7} +5 z^8 a^{-6} -17 z^6 a^{-6} +18 z^4 a^{-6} -10 z^2 a^{-6} +3 a^{-6} +4 z^9 a^{-5} -10 z^7 a^{-5} +5 z^5 a^{-5} +z^3 a^{-5} +z a^{-5} - a^{-5} z^{-1} +z^{10} a^{-4} +8 z^8 a^{-4} -38 z^6 a^{-4} +58 z^4 a^{-4} -34 z^2 a^{-4} +7 a^{-4} +7 z^9 a^{-3} -21 z^7 a^{-3} +a^3 z^5+30 z^5 a^{-3} -3 a^3 z^3-21 z^3 a^{-3} +3 a^3 z+9 z a^{-3} -a^3 z^{-1} -2 a^{-3} z^{-1} +z^{10} a^{-2} +6 z^8 a^{-2} +2 a^2 z^6-23 z^6 a^{-2} -3 a^2 z^4+37 z^4 a^{-2} -23 z^2 a^{-2} +a^2+4 a^{-2} +3 z^9 a^{-1} +3 a z^7-5 z^7 a^{-1} -3 a z^5+12 z^5 a^{-1} -2 a z^3-16 z^3 a^{-1} +3 a z+7 z a^{-1} -a z^{-1} - a^{-1} z^{-1} +3 z^8-z^6-3 z^4$ (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 16                       1 -1 14                     2   2 12                   4 1   -3 10                 5 2     3 8               8 4       -4 6             7 5         2 4           7 8           1 2         6 7             -1 0       4 8               4 -2     2 5                 -3 -4     4                   4 -6 1 2                     -1 -8 1                       1

Integral Khovanov Homology (db, data source)

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

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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