L10a93

L10a92

L10a94

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Link Presentations

[edit Notes on L10a93'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 X14,5,15,6 X18,13,19,14 X16,7,17,8 X6,15,7,16 X8,17,1,18
Gauss code	{1, -4, 2, -5, 6, -9, 8, -10}, {4, -1, 3, -2, 7, -6, 9, -8, 10, -7, 5, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{u^3 v^3-2 u^3 v^2+u^3 v-2 u^2 v^3+5 u^2 v^2-3 u^2 v+u^2+u v^3-3 u v^2+5 u v-2 u+v^2-2 v+1}{u^{3/2} v^{3/2}}$ (db)
Jones polynomial	$q^{3/2}-3 \sqrt{q}+\frac{4}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{9}{q^{5/2}}-\frac{10}{q^{7/2}}+\frac{9}{q^{9/2}}-\frac{8}{q^{11/2}}+\frac{5}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{1}{q^{17/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-z^3 a^7-2 z a^7+2 z^5 a^5+7 z^3 a^5+6 z a^5+a^5 z^{-1} -z^7 a^3-5 z^5 a^3-9 z^3 a^3-7 z a^3-a^3 z^{-1} +z^5 a+3 z^3 a+z a$ (db)
Kauffman polynomial	$-z^4 a^{10}+z^2 a^{10}-3 z^5 a^9+4 z^3 a^9-z a^9-4 z^6 a^8+4 z^4 a^8-4 z^7 a^7+4 z^5 a^7-z^3 a^7-3 z^8 a^6+3 z^6 a^6-2 z^4 a^6+z^2 a^6-z^9 a^5-5 z^7 a^5+18 z^5 a^5-21 z^3 a^5+9 z a^5-a^5 z^{-1} -6 z^8 a^4+17 z^6 a^4-16 z^4 a^4+4 z^2 a^4+a^4-z^9 a^3-4 z^7 a^3+22 z^5 a^3-26 z^3 a^3+10 z a^3-a^3 z^{-1} -3 z^8 a^2+9 z^6 a^2-6 z^4 a^2+z^2 a^2-3 z^7 a+11 z^5 a-10 z^3 a+2 z a-z^6+3 z^4-z^2$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                     1 -1 2                   2   2 0                 2 1   -1 -2               5 2     3 -4             5 3       -2 -6           5 4         1 -8         4 5           1 -10       4 5             -1 -12     2 5               3 -14   1 3                 -2 -16   2                   2 -18 1                     -1

Integral Khovanov Homology (db, data source)

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

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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