L10a23

L10a22

L10a24

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Link Presentations

[edit Notes on L10a23's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{(u-1) (v-1) \left(v^2+1\right) \left(v^2-3 v+1\right)}{\sqrt{u} v^{5/2}}$ (db)
Jones polynomial	$q^{3/2}-4 \sqrt{q}+\frac{6}{\sqrt{q}}-\frac{10}{q^{3/2}}+\frac{12}{q^{5/2}}-\frac{14}{q^{7/2}}+\frac{12}{q^{9/2}}-\frac{10}{q^{11/2}}+\frac{7}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{1}{q^{17/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^7 \left(-z^3\right)-2 a^7 z-a^7 z^{-1} +2 a^5 z^5+6 a^5 z^3+5 a^5 z+2 a^5 z^{-1} -a^3 z^7-4 a^3 z^5-5 a^3 z^3-2 a^3 z+a z^5+2 a z^3-a z-a z^{-1}$ (db)
Kauffman polynomial	$a^{10} z^4-a^{10} z^2+3 a^9 z^5-2 a^9 z^3+6 a^8 z^6-8 a^8 z^4+6 a^8 z^2-2 a^8+7 a^7 z^7-10 a^7 z^5+8 a^7 z^3-4 a^7 z+a^7 z^{-1} +5 a^6 z^8-2 a^6 z^6-9 a^6 z^4+12 a^6 z^2-5 a^6+2 a^5 z^9+7 a^5 z^7-23 a^5 z^5+20 a^5 z^3-9 a^5 z+2 a^5 z^{-1} +10 a^4 z^8-23 a^4 z^6+12 a^4 z^4+3 a^4 z^2-3 a^4+2 a^3 z^9+4 a^3 z^7-22 a^3 z^5+18 a^3 z^3-4 a^3 z+5 a^2 z^8-14 a^2 z^6+10 a^2 z^4-2 a^2 z^2+a^2+4 a z^7-12 a z^5+8 a z^3+a z-a z^{-1} +z^6-2 z^4$ (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                   3   3 0                 3 1   -2 -2               7 3     4 -4             7 5       -2 -6           7 5         2 -8         5 7           2 -10       5 7             -2 -12     2 5               3 -14   1 5                 -4 -16   2                   2 -18 1                     -1

Integral Khovanov Homology (db, data source)

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

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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