L11a139

L11a138

L11a140

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Link Presentations

[edit Notes on L11a139's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X22,10,7,9 X2738 X4,15,5,16 X12,5,13,6 X16,12,17,11 X6,18,1,17 X14,20,15,19 X20,14,21,13 X18,21,19,22
Gauss code	{1, -4, 2, -5, 6, -8}, {4, -1, 3, -2, 7, -6, 10, -9, 5, -7, 8, -11, 9, -10, 11, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{u^2 v^4-5 u^2 v^3+8 u^2 v^2-5 u^2 v+u^2-2 u v^4+10 u v^3-15 u v^2+10 u v-2 u+v^4-5 v^3+8 v^2-5 v+1}{u v^2}$ (db)
Jones polynomial	$-q^{13/2}+4 q^{11/2}-9 q^{9/2}+16 q^{7/2}-22 q^{5/2}+25 q^{3/2}-26 \sqrt{q}+\frac{22}{\sqrt{q}}-\frac{17}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{5}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^7 a^{-1} +2 a z^5-3 z^5 a^{-1} +2 z^5 a^{-3} -a^3 z^3+3 a z^3-5 z^3 a^{-1} +4 z^3 a^{-3} -z^3 a^{-5} -3 z a^{-1} +3 z a^{-3} -z a^{-5} +a^3 z^{-1} -a z^{-1}$ (db)
Kauffman polynomial	$z^5 a^{-7} -z^3 a^{-7} +4 z^6 a^{-6} -5 z^4 a^{-6} +2 z^2 a^{-6} +8 z^7 a^{-5} -11 z^5 a^{-5} +7 z^3 a^{-5} -2 z a^{-5} +10 z^8 a^{-4} +a^4 z^6-12 z^6 a^{-4} -a^4 z^4+5 z^4 a^{-4} +7 z^9 a^{-3} +5 a^3 z^7+4 z^7 a^{-3} -10 a^3 z^5-25 z^5 a^{-3} +5 a^3 z^3+22 z^3 a^{-3} +a^3 z-6 z a^{-3} -a^3 z^{-1} +2 z^{10} a^{-2} +9 a^2 z^8+21 z^8 a^{-2} -19 a^2 z^6-49 z^6 a^{-2} +11 a^2 z^4+34 z^4 a^{-2} -2 a^2 z^2-8 z^2 a^{-2} +a^2+7 a z^9+14 z^9 a^{-1} -2 a z^7-11 z^7 a^{-1} -22 a z^5-25 z^5 a^{-1} +17 a z^3+26 z^3 a^{-1} -a z-6 z a^{-1} -a z^{-1} +2 z^{10}+20 z^8-53 z^6+36 z^4-8 z^2$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                       1 1 12                     3   -3 10                   6 1   5 8                 10 3     -7 6               12 6       6 4             13 10         -3 2           13 12           1 0         10 14             4 -2       7 12               -5 -4     4 11                 7 -6   1 6                   -5 -8   4                     4 -10 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=-1$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=0$ ${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{13}$ $r=1$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{13}$ $r=2$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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