L11n58

L11n57

L11n59

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Link Presentations

[edit Notes on L11n58's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X7,14,8,15 X18,11,19,12 X22,19,5,20 X20,15,21,16 X16,21,17,22 X12,17,13,18 X13,8,14,9 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, 4, -8, -9, 3, 6, -7, 8, -4, 5, -6, 7, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{t(2)^5+3 t(1) t(2)^4-3 t(2)^4-6 t(1) t(2)^3+6 t(2)^3+6 t(1) t(2)^2-6 t(2)^2-3 t(1) t(2)+3 t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}}$ (db)
Jones polynomial	$-\frac{8}{q^{9/2}}+\frac{3}{q^{7/2}}-\frac{1}{q^{5/2}}+\frac{2}{q^{23/2}}-\frac{5}{q^{21/2}}+\frac{9}{q^{19/2}}-\frac{12}{q^{17/2}}+\frac{13}{q^{15/2}}-\frac{13}{q^{13/2}}+\frac{10}{q^{11/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-2 z a^{11}-2 a^{11} z^{-1} +5 z^3 a^9+11 z a^9+5 a^9 z^{-1} -3 z^5 a^7-10 z^3 a^7-10 z a^7-3 a^7 z^{-1} -z^5 a^5-2 z^3 a^5-z a^5$ (db)
Kauffman polynomial	$-3 z^4 a^{14}+4 z^2 a^{14}-a^{14}-z^7 a^{13}-3 z^5 a^{13}+4 z^3 a^{13}-z a^{13}-2 z^8 a^{12}-z^6 a^{12}+z^4 a^{12}+z^2 a^{12}-z^9 a^{11}-6 z^7 a^{11}+10 z^5 a^{11}-6 z^3 a^{11}+4 z a^{11}-2 a^{11} z^{-1} -6 z^8 a^{10}+z^6 a^{10}+16 z^4 a^{10}-17 z^2 a^{10}+5 a^{10}-z^9 a^9-11 z^7 a^9+27 z^5 a^9-26 z^3 a^9+17 z a^9-5 a^9 z^{-1} -4 z^8 a^8-z^6 a^8+16 z^4 a^8-15 z^2 a^8+5 a^8-6 z^7 a^7+13 z^5 a^7-14 z^3 a^7+11 z a^7-3 a^7 z^{-1} -3 z^6 a^6+4 z^4 a^6-z^2 a^6-z^5 a^5+2 z^3 a^5-z a^5$ (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 χ -4                   1 1 -6                 3 1 -2 -8               5     5 -10             5 3     -2 -12           8 5       3 -14         6 6         0 -16       6 7           -1 -18     3 6             3 -20   2 6               -4 -22   3                 3 -24 2                   -2

Integral Khovanov Homology (db, data source)

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

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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