L11a312

L11a311

L11a313

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Link Presentations

[edit Notes on L11a312's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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