L11a482

L11a481

L11a483

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Link Presentations

[edit Notes on L11a482's Link Presentations]

Planar diagram presentation	X6172 X2,9,3,10 X12,3,13,4 X10,5,11,6 X16,11,5,12 X4,15,1,16 X20,14,21,13 X18,7,19,8 X8,17,9,18 X22,20,17,19 X14,22,15,21
Gauss code	{1, -2, 3, -6}, {9, -8, 10, -7, 11, -10}, {4, -1, 8, -9, 2, -4, 5, -3, 7, -11, 6, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(t(1)-1) (t(2)-1) (t(3)-2) (t(3)-1) (2 t(3)-1)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}}$ (db)
Jones polynomial	$-q^2+5 q-11+17 q^{-1} -20 q^{-2} +25 q^{-3} -22 q^{-4} +19 q^{-5} -13 q^{-6} +7 q^{-7} -3 q^{-8} + q^{-9}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$z^2 a^8+a^8-2 z^4 a^6-3 z^2 a^6+a^6 z^{-2} -2 a^6+z^6 a^4+z^4 a^4+z^2 a^4-2 a^4 z^{-2} -a^4+z^6 a^2+z^4 a^2+z^2 a^2+a^2 z^{-2} +2 a^2-z^4$ (db)
Kauffman polynomial	$z^6 a^{10}-3 z^4 a^{10}+3 z^2 a^{10}-a^{10}+3 z^7 a^9-8 z^5 a^9+7 z^3 a^9-2 z a^9+4 z^8 a^8-5 z^6 a^8-4 z^4 a^8+6 z^2 a^8-a^8+4 z^9 a^7-z^7 a^7-11 z^5 a^7+13 z^3 a^7-6 z a^7+2 z^{10} a^6+8 z^8 a^6-20 z^6 a^6+11 z^4 a^6+z^2 a^6+a^6 z^{-2} -a^6+12 z^9 a^5-17 z^7 a^5+z^5 a^5+8 z^3 a^5-2 z a^5-2 a^5 z^{-1} +2 z^{10} a^4+17 z^8 a^4-39 z^6 a^4+23 z^4 a^4-z^2 a^4+2 a^4 z^{-2} -2 a^4+8 z^9 a^3-2 z^7 a^3-12 z^5 a^3+4 z^3 a^3+2 z a^3-2 a^3 z^{-1} +13 z^8 a^2-20 z^6 a^2+7 z^4 a^2+z^2 a^2+a^2 z^{-2} -2 a^2+11 z^7 a-15 z^5 a+2 z^3 a+5 z^6-4 z^4+z^5 a^{-1}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 5                       1 -1 3                     4   4 1                   7 1   -6 -1                 10 4     6 -3               12 9       -3 -5             13 8         5 -7           9 12           3 -9         10 13             -3 -11       5 11               6 -13     2 8                 -6 -15   1 5                   4 -17   2                     -2 -19 1                       1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-3$ $i=-1$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-4$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{10}$ $r=-3$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=-2$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{13}$ $r=-1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=0$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{10}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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