L11n456

L11n455

L11n457

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Link Presentations

[edit Notes on L11n456's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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