L10a46

L10a45

L10a47

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Link Presentations

[edit Notes on L10a46's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{\left(t(2)^2-t(2)+1\right) \left(t(1) t(2)^3-2 t(2)^3-t(1) t(2)^2+t(2)^2+t(1) t(2)-t(2)-2 t(1)+1\right)}{\sqrt{t(1)} t(2)^{5/2}}$ (db)
Jones polynomial	$9 q^{9/2}-9 q^{7/2}+10 q^{5/2}-\frac{1}{q^{5/2}}-9 q^{3/2}+\frac{1}{q^{3/2}}-q^{15/2}+3 q^{13/2}-6 q^{11/2}+6 \sqrt{q}-\frac{5}{\sqrt{q}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^5 a^{-5} -3 z^3 a^{-5} -3 z a^{-5} - a^{-5} z^{-1} +z^7 a^{-3} +5 z^5 a^{-3} +10 z^3 a^{-3} +11 z a^{-3} +5 a^{-3} z^{-1} -2 z^5 a^{-1} +a z^3-9 z^3 a^{-1} +4 a z-14 z a^{-1} +4 a z^{-1} -8 a^{-1} z^{-1}$ (db)
Kauffman polynomial	$z^3 a^{-9} +3 z^4 a^{-8} +6 z^5 a^{-7} -4 z^3 a^{-7} +z a^{-7} +9 z^6 a^{-6} -15 z^4 a^{-6} +9 z^2 a^{-6} -2 a^{-6} +8 z^7 a^{-5} -14 z^5 a^{-5} +4 z^3 a^{-5} + a^{-5} z^{-1} +4 z^8 a^{-4} -22 z^4 a^{-4} +23 z^2 a^{-4} -9 a^{-4} +z^9 a^{-3} +7 z^7 a^{-3} -26 z^5 a^{-3} +20 z^3 a^{-3} -9 z a^{-3} +5 a^{-3} z^{-1} +5 z^8 a^{-2} -11 z^6 a^{-2} -9 z^4 a^{-2} +28 z^2 a^{-2} -14 a^{-2} +z^9 a^{-1} +a z^7-6 a z^5-12 z^5 a^{-1} +13 a z^3+24 z^3 a^{-1} -12 a z-20 z a^{-1} +4 a z^{-1} +8 a^{-1} z^{-1} +z^8-2 z^6-5 z^4+14 z^2-8$ (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 χ 16                     1 1 14                   2   -2 12                 4 1   3 10               5 2     -3 8             4 4       0 6           6 5         -1 4         3 4           -1 2       4 7             3 0     1 2               -1 -2     4                 4 -4 1 1                   0 -6 1                     1

Integral Khovanov Homology (db, data source)

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

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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