L11a309

L11a308

L11a310

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Link Presentations

[edit Notes on L11a309's Link Presentations]

Planar diagram presentation	X10,1,11,2 X18,11,19,12 X6,9,7,10 X20,7,21,8 X8,19,1,20 X22,15,9,16 X12,4,13,3 X16,6,17,5 X4,14,5,13 X14,21,15,22 X2,18,3,17
Gauss code	{1, -11, 7, -9, 8, -3, 4, -5}, {3, -1, 2, -7, 9, -10, 6, -8, 11, -2, 5, -4, 10, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(t(1)-1) (t(2)-1) \left(t(1) t(2)^4+t(1)^2 t(2)^3-3 t(1) t(2)^3+t(2)^3-t(1)^2 t(2)^2+5 t(1) t(2)^2-t(2)^2+t(1)^2 t(2)-3 t(1) t(2)+t(2)+t(1)\right)}{t(1)^{3/2} t(2)^{5/2}}$ (db)
Jones polynomial	$-q^{7/2}+4 q^{5/2}-9 q^{3/2}+15 \sqrt{q}-\frac{22}{\sqrt{q}}+\frac{24}{q^{3/2}}-\frac{25}{q^{5/2}}+\frac{21}{q^{7/2}}-\frac{16}{q^{9/2}}+\frac{10}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^5 z^5-2 a^5 z^3-2 a^5 z-a^5 z^{-1} +a^3 z^7+3 a^3 z^5+4 a^3 z^3+4 a^3 z+3 a^3 z^{-1} +a z^7+3 a z^5-z^5 a^{-1} +3 a z^3-2 z^3 a^{-1} -a z-z a^{-1} -2 a z^{-1}$ (db)
Kauffman polynomial	$a^8 z^6-2 a^8 z^4+a^8 z^2+4 a^7 z^7-8 a^7 z^5+4 a^7 z^3+8 a^6 z^8-18 a^6 z^6+13 a^6 z^4-6 a^6 z^2+a^6+8 a^5 z^9-14 a^5 z^7+7 a^5 z^5-7 a^5 z^3+4 a^5 z-a^5 z^{-1} +3 a^4 z^{10}+11 a^4 z^8-37 a^4 z^6+31 a^4 z^4-13 a^4 z^2+3 a^4+16 a^3 z^9-31 a^3 z^7+20 a^3 z^5+z^5 a^{-3} -10 a^3 z^3-z^3 a^{-3} +7 a^3 z-3 a^3 z^{-1} +3 a^2 z^{10}+13 a^2 z^8-36 a^2 z^6+4 z^6 a^{-2} +28 a^2 z^4-5 z^4 a^{-2} -8 a^2 z^2+z^2 a^{-2} +3 a^2+8 a z^9-5 a z^7+8 z^7 a^{-1} -8 a z^5-12 z^5 a^{-1} +9 a z^3+7 z^3 a^{-1} +a z-2 z a^{-1} -2 a z^{-1} +10 z^8-14 z^6+7 z^4-z^2$ (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 8                       1 1 6                     3   -3 4                   6 1   5 2                 9 3     -6 0               13 6       7 -2             13 11         -2 -4           12 11           1 -6         9 13             4 -8       7 12               -5 -10     3 9                 6 -12   1 7                   -6 -14   3                     3 -16 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-3$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=-2$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=-1$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{13}$ $r=0$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{13}$ $r=1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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