L10a123

L10a122

L10a124

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Link Presentations

[edit Notes on L10a123's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{t(1) t(3)^3+t(2) t(3)^3-t(3)^3-5 t(1) t(3)^2+2 t(1) t(2) t(3)^2-5 t(2) t(3)^2+4 t(3)^2+5 t(1) t(3)-4 t(1) t(2) t(3)+5 t(2) t(3)-2 t(3)-t(1)+t(1) t(2)-t(2)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}}$ (db)
Jones polynomial	$q^{-9} -2 q^{-8} +6 q^{-7} -9 q^{-6} +12 q^{-5} -12 q^{-4} +13 q^{-3} -10 q^{-2} +q+7 q^{-1} -3$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{10} z^{-2} -3 a^8 z^{-2} -4 a^8+6 a^6 z^2+4 a^6 z^{-2} +8 a^6-3 a^4 z^4-6 a^4 z^2-3 a^4 z^{-2} -7 a^4-a^2 z^4+2 a^2 z^2+a^2 z^{-2} +3 a^2+z^2$ (db)
Kauffman polynomial	$z^6 a^{10}-4 z^4 a^{10}+6 z^2 a^{10}+a^{10} z^{-2} -4 a^{10}+2 z^7 a^9-5 z^5 a^9+3 z^3 a^9+z a^9-a^9 z^{-1} +2 z^8 a^8+z^6 a^8-17 z^4 a^8+25 z^2 a^8+3 a^8 z^{-2} -14 a^8+z^9 a^7+6 z^7 a^7-17 z^5 a^7+7 z^3 a^7+z a^7-a^7 z^{-1} +6 z^8 a^6-z^6 a^6-29 z^4 a^6+38 z^2 a^6+4 a^6 z^{-2} -21 a^6+z^9 a^5+11 z^7 a^5-24 z^5 a^5+11 z^3 a^5+z a^5-a^5 z^{-1} +4 z^8 a^4+5 z^6 a^4-25 z^4 a^4+28 z^2 a^4+3 a^4 z^{-2} -14 a^4+7 z^7 a^3-9 z^5 a^3+5 z^3 a^3+z a^3-a^3 z^{-1} +6 z^6 a^2-8 z^4 a^2+8 z^2 a^2+a^2 z^{-2} -4 a^2+3 z^5 a-2 z^3 a+z^4-z^2$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 3                     1 1 1                   2   -2 -1                 5 1   4 -3               6 3     -3 -5             7 4       3 -7           5 6         1 -9         7 7           0 -11       5 8             3 -13     1 4               -3 -15   1 5                 4 -17   1                   -1 -19 1                     1

Integral Khovanov Homology (db, data source)

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

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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