L11a89

L11a88

L11a90

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Link Presentations

[edit Notes on L11a89's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X16,10,17,9 X22,13,5,14 X14,21,15,22 X10,16,11,15 X20,17,21,18 X18,7,19,8 X8,19,9,20 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 8, -9, 3, -6, 11, -2, 4, -5, 6, -3, 7, -8, 9, -7, 5, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{2 u v^4-8 u v^3+10 u v^2-6 u v+u+v^5-6 v^4+10 v^3-8 v^2+2 v}{\sqrt{u} v^{5/2}}$ (db)
Jones polynomial	$-\sqrt{q}+\frac{3}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{11}{q^{5/2}}-\frac{15}{q^{7/2}}+\frac{17}{q^{9/2}}-\frac{18}{q^{11/2}}+\frac{14}{q^{13/2}}-\frac{11}{q^{15/2}}+\frac{7}{q^{17/2}}-\frac{3}{q^{19/2}}+\frac{1}{q^{21/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{11} z^{-1} +4 z a^9+3 a^9 z^{-1} -5 z^3 a^7-8 z a^7-3 a^7 z^{-1} +2 z^5 a^5+4 z^3 a^5+5 z a^5+2 a^5 z^{-1} +z^5 a^3-2 z a^3-a^3 z^{-1} -z^3 a-z a$ (db)
Kauffman polynomial	$-z^6 a^{12}+3 z^4 a^{12}-3 z^2 a^{12}+a^{12}-3 z^7 a^{11}+8 z^5 a^{11}-6 z^3 a^{11}+3 z a^{11}-a^{11} z^{-1} -4 z^8 a^{10}+7 z^6 a^{10}+2 z^4 a^{10}-4 z^2 a^{10}+2 a^{10}-3 z^9 a^9-z^7 a^9+19 z^5 a^9-22 z^3 a^9+12 z a^9-3 a^9 z^{-1} -z^{10} a^8-9 z^8 a^8+28 z^6 a^8-26 z^4 a^8+9 z^2 a^8-7 z^9 a^7+8 z^7 a^7+14 z^5 a^7-32 z^3 a^7+16 z a^7-3 a^7 z^{-1} -z^{10} a^6-11 z^8 a^6+34 z^6 a^6-40 z^4 a^6+16 z^2 a^6-2 a^6-4 z^9 a^5+z^7 a^5+12 z^5 a^5-22 z^3 a^5+12 z a^5-2 a^5 z^{-1} -6 z^8 a^4+11 z^6 a^4-10 z^4 a^4+5 z^2 a^4-5 z^7 a^3+8 z^5 a^3-4 z^3 a^3+4 z a^3-a^3 z^{-1} -3 z^6 a^2+5 z^4 a^2-z^2 a^2-z^5 a+2 z^3 a-z a$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                       1 1 0                     2   -2 -2                   5 1   4 -4                 7 3     -4 -6               8 4       4 -8             9 7         -2 -10           9 8           1 -12         6 10             4 -14       5 8               -3 -16     2 6                 4 -18   1 5                   -4 -20   2                     2 -22 1                       -1

Integral Khovanov Homology (db, data source)

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

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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