L10n58

L10n57

L10n59

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Link Presentations

[edit Notes on L10n58's Link Presentations]

Planar diagram presentation	X12,1,13,2 X7,17,8,16 X5,1,6,10 X3746 X9,5,10,4 X17,11,18,20 X13,19,14,18 X19,15,20,14 X2,11,3,12 X15,9,16,8
Gauss code	{1, -9, -4, 5, -3, 4, -2, 10, -5, 3}, {9, -1, -7, 8, -10, 2, -6, 7, -8, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{(u-1) (v-1) \left(u^2 v+u v^2-2 u v+u+v\right)}{u^{3/2} v^{3/2}}$ (db)
Jones polynomial	$8 q^{9/2}-8 q^{7/2}+4 q^{5/2}-3 q^{3/2}-q^{19/2}+3 q^{17/2}-5 q^{15/2}+8 q^{13/2}-8 q^{11/2}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z a^{-9} - a^{-9} z^{-1} +3 z^3 a^{-7} +7 z a^{-7} +5 a^{-7} z^{-1} -2 z^5 a^{-5} -8 z^3 a^{-5} -13 z a^{-5} -8 a^{-5} z^{-1} +3 z^3 a^{-3} +7 z a^{-3} +4 a^{-3} z^{-1}$ (db)
Kauffman polynomial	$z^5 a^{-11} -2 z^3 a^{-11} +z a^{-11} +3 z^6 a^{-10} -7 z^4 a^{-10} +5 z^2 a^{-10} -2 a^{-10} +3 z^7 a^{-9} -3 z^5 a^{-9} -4 z^3 a^{-9} +z a^{-9} + a^{-9} z^{-1} +z^8 a^{-8} +8 z^6 a^{-8} -26 z^4 a^{-8} +24 z^2 a^{-8} -9 a^{-8} +7 z^7 a^{-7} -14 z^5 a^{-7} +12 z^3 a^{-7} -9 z a^{-7} +5 a^{-7} z^{-1} +z^8 a^{-6} +8 z^6 a^{-6} -25 z^4 a^{-6} +31 z^2 a^{-6} -14 a^{-6} +4 z^7 a^{-5} -10 z^5 a^{-5} +20 z^3 a^{-5} -19 z a^{-5} +8 a^{-5} z^{-1} +3 z^6 a^{-4} -6 z^4 a^{-4} +12 z^2 a^{-4} -8 a^{-4} +6 z^3 a^{-3} -10 z a^{-3} +4 a^{-3} z^{-1}$ (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   \ 0 1 2 3 4 5 6 7 8 χ 20                 1 1 18               2   -2 16             3 1   2 14           5 2     -3 12         3 3       0 10       5 5         0 8     3 3           0 6   1 5             4 4 2 3               -1 2 3                 3

Integral Khovanov Homology (db, data source)

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

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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