L11a526

L11a525

L11a527

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Link Presentations

[edit Notes on L11a526's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(t(3)-1) \left(-t(2)^2 t(1)^2+t(2) t(3)^2 t(1)^2-t(3)^2 t(1)^2+2 t(2) t(1)^2-2 t(2) t(3) t(1)^2+t(3) t(1)^2-t(1)^2+2 t(2)^2 t(1)+t(2)^2 t(3)^2 t(1)-3 t(2) t(3)^2 t(1)+2 t(3)^2 t(1)-3 t(2) t(1)-2 t(2)^2 t(3) t(1)+6 t(2) t(3) t(1)-2 t(3) t(1)+t(1)-t(2)^2-t(2)^2 t(3)^2+2 t(2) t(3)^2-t(3)^2+t(2)+t(2)^2 t(3)-2 t(2) t(3)\right)}{t(1) t(2) t(3)^{3/2}}$ (db)
Jones polynomial	$q^3-4 q^2+10 q-16+23 q^{-1} -25 q^{-2} +27 q^{-3} -22 q^{-4} +17 q^{-5} -10 q^{-6} +4 q^{-7} - q^{-8}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^6 \left(-z^4\right)-a^6 z^2-2 a^6+a^4 z^6+2 a^4 z^4+6 a^4 z^2+a^4 z^{-2} +6 a^4+a^2 z^6-a^2 z^4-6 a^2 z^2-2 a^2 z^{-2} +z^2 a^{-2} -7 a^2-2 z^4+ z^{-2} +3$ (db)
Kauffman polynomial	$a^9 z^5-a^9 z^3+4 a^8 z^6-4 a^8 z^4+9 a^7 z^7-13 a^7 z^5+7 a^7 z^3-2 a^7 z+13 a^6 z^8-25 a^6 z^6+24 a^6 z^4-13 a^6 z^2+4 a^6+10 a^5 z^9-10 a^5 z^7-5 a^5 z^5+13 a^5 z^3-6 a^5 z+3 a^4 z^{10}+20 a^4 z^8-64 a^4 z^6+73 a^4 z^4-42 a^4 z^2-a^4 z^{-2} +13 a^4+18 a^3 z^9-35 a^3 z^7+18 a^3 z^5+3 a^3 z^3-7 a^3 z+2 a^3 z^{-1} +3 a^2 z^{10}+15 a^2 z^8-54 a^2 z^6+z^6 a^{-2} +62 a^2 z^4-2 z^4 a^{-2} -40 a^2 z^2+z^2 a^{-2} -2 a^2 z^{-2} +13 a^2+8 a z^9-12 a z^7+4 z^7 a^{-1} +a z^5-8 z^5 a^{-1} +2 a z^3+4 z^3 a^{-1} -3 a z+2 a z^{-1} +8 z^8-18 z^6+15 z^4-10 z^2- z^{-2} +5$ (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 7                       1 1 5                     3   -3 3                   7 1   6 1                 9 3     -6 -1               14 7       7 -3             13 11         -2 -5           14 12           2 -7         10 15             5 -9       7 12               -5 -11     3 10                 7 -13   1 7                   -6 -15   3                     3 -17 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-3$ $i=-1$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-3$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=-2$ ${\mathbb Z}^{15}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{14}$ $r=-1$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{13}$ $r=0$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{14}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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