L10a91

L10a90

L10a92

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Link Presentations

[edit Notes on L10a91's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,4,13,3 X20,12,9,11 X2,9,3,10 X4,20,5,19 X14,6,15,5 X16,7,17,8 X18,16,19,15 X6,17,7,18 X8,14,1,13
Gauss code	{1, -4, 2, -5, 6, -9, 7, -10}, {4, -1, 3, -2, 10, -6, 8, -7, 9, -8, 5, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(u-1) (v-1) \left(u^2 v-u^2+u v^2-4 u v+u-v^2+v\right)}{u^{3/2} v^{3/2}}$ (db)
Jones polynomial	$-q^{13/2}+4 q^{11/2}-7 q^{9/2}+10 q^{7/2}-13 q^{5/2}+13 q^{3/2}-13 \sqrt{q}+\frac{9}{\sqrt{q}}-\frac{6}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{1}{q^{7/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^3 a^{-5} +z^5 a^{-3} +z^3 a^{-3} +a^3 z+z a^{-3} +z^5 a^{-1} -2 a z^3-a z-z a^{-1} +a z^{-1} - a^{-1} z^{-1}$ (db)
Kauffman polynomial	$z^5 a^{-7} -z^3 a^{-7} +4 z^6 a^{-6} -7 z^4 a^{-6} +2 z^2 a^{-6} +6 z^7 a^{-5} -11 z^5 a^{-5} +5 z^3 a^{-5} -z a^{-5} +4 z^8 a^{-4} -11 z^4 a^{-4} +7 z^2 a^{-4} +z^9 a^{-3} +11 z^7 a^{-3} +a^3 z^5-26 z^5 a^{-3} -2 a^3 z^3+20 z^3 a^{-3} +a^3 z-6 z a^{-3} +7 z^8 a^{-2} +3 a^2 z^6-6 z^6 a^{-2} -6 a^2 z^4-6 z^4 a^{-2} +3 a^2 z^2+7 z^2 a^{-2} +z^9 a^{-1} +4 a z^7+9 z^7 a^{-1} -6 a z^5-21 z^5 a^{-1} +3 a z^3+19 z^3 a^{-1} -2 a z-8 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +3 z^8+z^6-8 z^4+5 z^2-1$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                     1 1 12                   3   -3 10                 4 1   3 8               6 3     -3 6             7 4       3 4           6 6         0 2         7 7           0 0       4 8             4 -2     2 5               -3 -4   1 4                 3 -6   2                   -2 -8 1                     1

Integral Khovanov Homology (db, data source)

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

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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