L11a16

L11a15

L11a17

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Link Presentations

[edit Notes on L11a16's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X14,6,15,5 X8493 X18,10,19,9 X20,12,21,11 X22,14,5,13 X10,20,11,19 X12,22,13,21 X2,16,3,15
Gauss code	{1, -11, 5, -3}, {4, -1, 2, -5, 6, -9, 7, -10, 8, -4, 11, -2, 3, -6, 9, -7, 10, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(t(1)-1) (t(2)-1) \left(t(2)^6-t(2)^5+t(2)^4-t(2)^3+t(2)^2-t(2)+1\right)}{\sqrt{t(1)} t(2)^{7/2}}$ (db)
Jones polynomial	$-8 q^{9/2}+6 q^{7/2}-6 q^{5/2}+3 q^{3/2}-q^{21/2}+3 q^{19/2}-4 q^{17/2}+6 q^{15/2}-7 q^{13/2}+8 q^{11/2}-3 \sqrt{q}+\frac{1}{\sqrt{q}}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$z^9 a^{-5} -z^7 a^{-3} +7 z^7 a^{-5} -z^7 a^{-7} -5 z^5 a^{-3} +16 z^5 a^{-5} -5 z^5 a^{-7} -5 z^3 a^{-3} +12 z^3 a^{-5} -6 z^3 a^{-7} +3 z a^{-3} -3 z a^{-5} +3 a^{-3} z^{-1} -5 a^{-5} z^{-1} +2 a^{-7} z^{-1}$ (db)
Kauffman polynomial	$z^3 a^{-13} +3 z^4 a^{-12} -2 z^2 a^{-12} +4 z^5 a^{-11} -3 z^3 a^{-11} +4 z^6 a^{-10} -3 z^4 a^{-10} -2 z^2 a^{-10} + a^{-10} +4 z^7 a^{-9} -6 z^5 a^{-9} +4 z^8 a^{-8} -10 z^6 a^{-8} +4 z^4 a^{-8} +4 z^9 a^{-7} -16 z^7 a^{-7} +20 z^5 a^{-7} -12 z^3 a^{-7} +z a^{-7} +2 a^{-7} z^{-1} +2 z^{10} a^{-6} -6 z^8 a^{-6} -z^6 a^{-6} +6 z^4 a^{-6} +4 z^2 a^{-6} -5 a^{-6} +7 z^9 a^{-5} -38 z^7 a^{-5} +63 z^5 a^{-5} -35 z^3 a^{-5} -z a^{-5} +5 a^{-5} z^{-1} +2 z^{10} a^{-4} -9 z^8 a^{-4} +8 z^6 a^{-4} +2 z^4 a^{-4} +3 z^2 a^{-4} -5 a^{-4} +3 z^9 a^{-3} -18 z^7 a^{-3} +33 z^5 a^{-3} -19 z^3 a^{-3} -2 z a^{-3} +3 a^{-3} z^{-1} +z^8 a^{-2} -5 z^6 a^{-2} +6 z^4 a^{-2} -z^2 a^{-2}$ (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   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 22                       1 1 20                     2   -2 18                   2 1   1 16                 4 2     -2 14               3 2       1 12             5 4         -1 10           3 3           0 8         3 5             2 6       3 3               0 4     2 5                 3 2   1 1                   0 0   2                     2 -2 1                       -1

Integral Khovanov Homology (db, data source)

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

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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