L10a45

L10a44

L10a46

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Link Presentations

[edit Notes on L10a45's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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