L11a27

L11a26

L11a28

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Link Presentations

[edit Notes on L11a27's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{t(2)^5+2 t(1) t(2)^4-6 t(2)^4-9 t(1) t(2)^3+11 t(2)^3+11 t(1) t(2)^2-9 t(2)^2-6 t(1) t(2)+2 t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}}$ (db)
Jones polynomial	$-\sqrt{q}+\frac{3}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{12}{q^{5/2}}-\frac{17}{q^{7/2}}+\frac{18}{q^{9/2}}-\frac{19}{q^{11/2}}+\frac{16}{q^{13/2}}-\frac{12}{q^{15/2}}+\frac{7}{q^{17/2}}-\frac{3}{q^{19/2}}+\frac{1}{q^{21/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{11} z^{-1} +4 z a^9+3 a^9 z^{-1} -5 z^3 a^7-7 z a^7-3 a^7 z^{-1} +2 z^5 a^5+3 z^3 a^5+3 z a^5+2 a^5 z^{-1} +z^5 a^3-z a^3-a^3 z^{-1} -z^3 a-z a$ (db)
Kauffman polynomial	$a^{12} z^6-3 a^{12} z^4+3 a^{12} z^2-a^{12}+3 a^{11} z^7-8 a^{11} z^5+7 a^{11} z^3-3 a^{11} z+a^{11} z^{-1} +4 a^{10} z^8-6 a^{10} z^6-3 a^{10} z^4+6 a^{10} z^2-2 a^{10}+3 a^9 z^9+3 a^9 z^7-22 a^9 z^5+24 a^9 z^3-13 a^9 z+3 a^9 z^{-1} +a^8 z^{10}+10 a^8 z^8-25 a^8 z^6+17 a^8 z^4-5 a^8 z^2+7 a^7 z^9-3 a^7 z^7-21 a^7 z^5+31 a^7 z^3-17 a^7 z+3 a^7 z^{-1} +a^6 z^{10}+12 a^6 z^8-29 a^6 z^6+27 a^6 z^4-11 a^6 z^2+2 a^6+4 a^5 z^9+2 a^5 z^7-15 a^5 z^5+20 a^5 z^3-11 a^5 z+2 a^5 z^{-1} +6 a^4 z^8-8 a^4 z^6+5 a^4 z^4-a^4 z^2+5 a^3 z^7-7 a^3 z^5+4 a^3 z^3-3 a^3 z+a^3 z^{-1} +3 a^2 z^6-5 a^2 z^4+2 a^2 z^2+a z^5-2 a z^3+a z$ (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                 8 3     -5 -6               9 4       5 -8             9 8         -1 -10           10 9           1 -12         7 10             3 -14       5 9               -4 -16     2 7                 5 -18   1 5                   -4 -20   2                     2 -22 1                       -1

Integral Khovanov Homology (db, data source)

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

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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