L11a17

L11a16

L11a18

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Link Presentations

[edit Notes on L11a17's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(t(1)-1) (t(2)-1) \left(2 t(2)^4-3 t(2)^3+3 t(2)^2-3 t(2)+2\right)}{\sqrt{t(1)} t(2)^{5/2}}$ (db)
Jones polynomial	$-q^{19/2}+4 q^{17/2}-7 q^{15/2}+12 q^{13/2}-15 q^{11/2}+16 q^{9/2}-17 q^{7/2}+13 q^{5/2}-10 q^{3/2}+5 \sqrt{q}-\frac{3}{\sqrt{q}}+\frac{1}{q^{3/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^7 a^{-3} +z^7 a^{-5} -z^5 a^{-1} +4 z^5 a^{-3} +3 z^5 a^{-5} -z^5 a^{-7} -3 z^3 a^{-1} +6 z^3 a^{-3} -2 z^3 a^{-7} -z a^{-1} +6 z a^{-3} -6 z a^{-5} +z a^{-7} +3 a^{-3} z^{-1} -5 a^{-5} z^{-1} +2 a^{-7} z^{-1}$ (db)
Kauffman polynomial	$-2 z^{10} a^{-4} -2 z^{10} a^{-6} -4 z^9 a^{-3} -9 z^9 a^{-5} -5 z^9 a^{-7} -4 z^8 a^{-2} -2 z^8 a^{-6} -6 z^8 a^{-8} -3 z^7 a^{-1} +10 z^7 a^{-3} +28 z^7 a^{-5} +9 z^7 a^{-7} -6 z^7 a^{-9} +11 z^6 a^{-2} +10 z^6 a^{-4} +9 z^6 a^{-6} +7 z^6 a^{-8} -4 z^6 a^{-10} -z^6+10 z^5 a^{-1} -5 z^5 a^{-3} -37 z^5 a^{-5} -12 z^5 a^{-7} +9 z^5 a^{-9} -z^5 a^{-11} -6 z^4 a^{-2} -13 z^4 a^{-4} -9 z^4 a^{-6} +2 z^4 a^{-8} +7 z^4 a^{-10} +3 z^4-8 z^3 a^{-1} -6 z^3 a^{-3} +13 z^3 a^{-5} +9 z^3 a^{-7} -z^3 a^{-9} +z^3 a^{-11} -2 z^2 a^{-4} -3 z^2 a^{-6} -2 z^2 a^{-8} -2 z^2 a^{-10} -z^2+2 z a^{-1} +8 z a^{-3} +7 z a^{-5} +z a^{-7} +5 a^{-4} +5 a^{-6} - a^{-10} -3 a^{-3} z^{-1} -5 a^{-5} z^{-1} -2 a^{-7} z^{-1}$ (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   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 20                       1 1 18                     3   -3 16                   4 1   3 14                 8 3     -5 12               7 4       3 10             9 8         -1 8           8 7           1 6         5 9             4 4       5 8               -3 2     2 7                 5 0   1 3                   -2 -2   2                     2 -4 1                       -1

Integral Khovanov Homology (db, data source)

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

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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