L11a304

L11a303

L11a305

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Link Presentations

[edit Notes on L11a304's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{(u v+1) \left(u^2 v^4-2 u^2 v^3+2 u^2 v^2+u v^3-2 u v^2+u v+2 v^2-2 v+1\right)}{u^{3/2} v^{5/2}}$ (db)
Jones polynomial	$\sqrt{q}-\frac{2}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{6}{q^{5/2}}+\frac{7}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{8}{q^{11/2}}-\frac{8}{q^{13/2}}+\frac{6}{q^{15/2}}-\frac{4}{q^{17/2}}+\frac{2}{q^{19/2}}-\frac{1}{q^{21/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^7 z^7+6 a^7 z^5+12 a^7 z^3+9 a^7 z+2 a^7 z^{-1} -a^5 z^9-8 a^5 z^7-24 a^5 z^5-33 a^5 z^3-19 a^5 z-3 a^5 z^{-1} +a^3 z^7+6 a^3 z^5+11 a^3 z^3+6 a^3 z+a^3 z^{-1}$ (db)
Kauffman polynomial	$a^{13} z^3-a^{13} z+2 a^{12} z^4-a^{12} z^2+3 a^{11} z^5-2 a^{11} z^3+a^{11} z+4 a^{10} z^6-5 a^{10} z^4+2 a^{10} z^2+5 a^9 z^7-12 a^9 z^5+10 a^9 z^3-3 a^9 z+4 a^8 z^8-9 a^8 z^6+2 a^8 z^4+a^8 z^2+3 a^7 z^9-9 a^7 z^7+9 a^7 z^5-13 a^7 z^3+10 a^7 z-2 a^7 z^{-1} +a^6 z^{10}+a^6 z^8-16 a^6 z^6+23 a^6 z^4-13 a^6 z^2+3 a^6+5 a^5 z^9-26 a^5 z^7+48 a^5 z^5-45 a^5 z^3+21 a^5 z-3 a^5 z^{-1} +a^4 z^{10}-2 a^4 z^8-9 a^4 z^6+25 a^4 z^4-17 a^4 z^2+3 a^4+2 a^3 z^9-12 a^3 z^7+24 a^3 z^5-19 a^3 z^3+6 a^3 z-a^3 z^{-1} +a^2 z^8-6 a^2 z^6+11 a^2 z^4-6 a^2 z^2+a^2$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 2                       1 -1 0                     1   1 -2                   2 1   -1 -4                 4 1     3 -6               4 3       -1 -8             4 3         1 -10           4 4           0 -12         4 4             0 -14       2 4               2 -16     2 4                 -2 -18     2                   2 -20 1 2                     -1 -22 1                       1

Integral Khovanov Homology (db, data source)

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

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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