L10a136

L10a135

L10a137

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Link Presentations

[edit Notes on L10a136's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X20,7,11,8 X8,19,9,20 X18,12,19,11 X16,13,17,14 X14,6,15,5 X10,16,5,15 X2,9,3,10 X4,18,1,17
Gauss code	{1, -9, 2, -10}, {7, -1, 3, -4, 9, -8}, {5, -2, 6, -7, 8, -6, 10, -5, 4, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=0$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{10}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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