L11a389

L11a388

L11a390

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Link Presentations

[edit Notes on L11a389's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X20,12,21,11 X22,17,9,18 X18,21,19,22 X16,14,17,13 X8,16,5,15 X14,8,15,7 X12,20,13,19 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 8, -7}, {11, -2, 3, -9, 6, -8, 7, -6, 4, -5, 9, -3, 5, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{-t(1) t(3)^3+t(1) t(2) t(3)^3-2 t(2) t(3)^3+t(3)^3+6 t(1) t(3)^2-5 t(1) t(2) t(3)^2+8 t(2) t(3)^2-5 t(3)^2-8 t(1) t(3)+5 t(1) t(2) t(3)-6 t(2) t(3)+5 t(3)+2 t(1)-t(1) t(2)+t(2)-1}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}}$ (db)
Jones polynomial	$-q^5+4 q^4-8 q^3+14 q^2-17 q+19-18 q^{-1} +16 q^{-2} -10 q^{-3} +6 q^{-4} -2 q^{-5} + q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^6 z^{-2} +a^6-3 z^2 a^4-3 a^4 z^{-2} -5 a^4+3 z^4 a^2+6 z^2 a^2+4 a^2 z^{-2} +8 a^2-z^6-2 z^4-5 z^2-3 z^{-2} -6+2 z^4 a^{-2} +2 z^2 a^{-2} + a^{-2} z^{-2} +2 a^{-2} -z^2 a^{-4}$ (db)
Kauffman polynomial	$a^2 z^{10}+z^{10}+2 a^3 z^9+6 a z^9+4 z^9 a^{-1} +2 a^4 z^8+5 a^2 z^8+7 z^8 a^{-2} +10 z^8+2 a^5 z^7+3 a^3 z^7-a z^7+5 z^7 a^{-1} +7 z^7 a^{-3} +a^6 z^6+2 a^4 z^6-a^2 z^6-5 z^6 a^{-2} +4 z^6 a^{-4} -11 z^6-5 a^5 z^5-12 a^3 z^5-11 a z^5-15 z^5 a^{-1} -10 z^5 a^{-3} +z^5 a^{-5} -4 a^6 z^4-18 a^4 z^4-26 a^2 z^4-7 z^4 a^{-2} -6 z^4 a^{-4} -13 z^4+3 a^5 z^3+8 a^3 z^3+5 a z^3+5 z^3 a^{-1} +4 z^3 a^{-3} -z^3 a^{-5} +6 a^6 z^2+25 a^4 z^2+38 a^2 z^2+8 z^2 a^{-2} +3 z^2 a^{-4} +24 z^2+a^5 z+a^3 z+a z+z a^{-1} -4 a^6-14 a^4-21 a^2-4 a^{-2} -14-a^5 z^{-1} -a^3 z^{-1} -a z^{-1} - a^{-1} z^{-1} +a^6 z^{-2} +3 a^4 z^{-2} +4 a^2 z^{-2} + a^{-2} z^{-2} +3 z^{-2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     3   3 7                   5 1   -4 5                 9 3     6 3               8 5       -3 1             11 9         2 -1           10 11           1 -3         6 8             -2 -5       4 10               6 -7     2 6                 -4 -9   1 5                   4 -11   1                     -1 -13 1                       1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=0$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{11}$ $r=1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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