L11n313

L11n312

L11n314

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Link Presentations

[edit Notes on L11n313's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X9,20,10,21 X7,16,8,17 X13,18,14,19 X19,14,20,15 X15,22,16,11 X17,10,18,5 X21,8,22,9 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 9, -3, 8}, {11, -2, -5, 6, -7, 4, -8, 5, -6, 3, -9, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{t(1) t(2) t(3)^4+t(1) t(2)^2 t(3)^3+t(2)^2 t(3)^3-t(1) t(3)^3-t(1) t(2) t(3)^3-t(2) t(3)^3-2 t(2)^2 t(3)^2+2 t(1) t(3)^2-t(1) t(2) t(3)^2+t(2) t(3)^2+t(2)^2 t(3)-t(1) t(3)+t(1) t(2) t(3)+t(2) t(3)-t(3)-t(2)}{\sqrt{t(1)} t(2) t(3)^2}$ (db)
Jones polynomial	$q^{-3} - q^{-4} + q^{-5} + q^{-6} +3 q^{-8} -2 q^{-9} +3 q^{-10} -3 q^{-11} +2 q^{-12} - q^{-13}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{14} z^{-2} +4 a^{12} z^{-2} +5 a^{12}-a^{10} z^4-9 a^{10} z^2-5 a^{10} z^{-2} -13 a^{10}+a^8 z^6+6 a^8 z^4+9 a^8 z^2+2 a^8 z^{-2} +7 a^8+a^6 z^6+5 a^6 z^4+5 a^6 z^2+a^6$ (db)
Kauffman polynomial	$z^7 a^{15}-5 z^5 a^{15}+7 z^3 a^{15}-4 z a^{15}+a^{15} z^{-1} +2 z^8 a^{14}-10 z^6 a^{14}+13 z^4 a^{14}-7 z^2 a^{14}-a^{14} z^{-2} +3 a^{14}+z^9 a^{13}-2 z^7 a^{13}-11 z^5 a^{13}+26 z^3 a^{13}-19 z a^{13}+5 a^{13} z^{-1} +4 z^8 a^{12}-23 z^6 a^{12}+38 z^4 a^{12}-29 z^2 a^{12}-4 a^{12} z^{-2} +15 a^{12}+z^9 a^{11}-2 z^7 a^{11}-16 z^5 a^{11}+42 z^3 a^{11}-33 z a^{11}+9 a^{11} z^{-1} +3 z^8 a^{10}-21 z^6 a^{10}+44 z^4 a^{10}-42 z^2 a^{10}-5 a^{10} z^{-2} +20 a^{10}+2 z^7 a^9-15 z^5 a^9+27 z^3 a^9-18 z a^9+5 a^9 z^{-1} +z^8 a^8-7 z^6 a^8+14 z^4 a^8-15 z^2 a^8-2 a^8 z^{-2} +8 a^8+z^7 a^7-5 z^5 a^7+4 z^3 a^7+z^6 a^6-5 z^4 a^6+5 z^2 a^6-a^6$ (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -5                       1 1 -7                     1 1 0 -9                 1 1     0 -11               2 1 1     2 -13             2 4 1       1 -15           3 1 1         3 -17         2 5 2           1 -19       2 2 1             1 -21     1 2 1               0 -23   1 2                   -1 -25   1                     1 -27 1                       -1

Integral Khovanov Homology (db, data source)

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

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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