L11a475

L11a474

L11a476

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Link Presentations

[edit Notes on L11a475's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{(t(2)-1) (t(3)-1)^2 (t(3) t(1)-3 t(1)-3 t(3)+1)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}}$ (db)
Jones polynomial	$-q^5+4 q^4-9 q^3+15 q^2-19 q+21-20 q^{-1} +18 q^{-2} -11 q^{-3} +7 q^{-4} -2 q^{-5} + q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^6 z^{-2} +a^6-3 a^4 z^2-2 a^4 z^{-2} -z^2 a^{-4} -4 a^4+3 a^2 z^4+2 z^4 a^{-2} +4 a^2 z^2+a^2 z^{-2} +z^2 a^{-2} +3 a^2-z^6-z^4-z^2$ (db)
Kauffman polynomial	$a^2 z^{10}+z^{10}+3 a^3 z^9+8 a z^9+5 z^9 a^{-1} +3 a^4 z^8+11 a^2 z^8+9 z^8 a^{-2} +17 z^8+2 a^5 z^7-3 a z^7+7 z^7 a^{-1} +8 z^7 a^{-3} +a^6 z^6-3 a^4 z^6-27 a^2 z^6-11 z^6 a^{-2} +4 z^6 a^{-4} -38 z^6-4 a^5 z^5-7 a^3 z^5-18 a z^5-28 z^5 a^{-1} -12 z^5 a^{-3} +z^5 a^{-5} -4 a^6 z^4-5 a^4 z^4+22 a^2 z^4+3 z^4 a^{-2} -5 z^4 a^{-4} +31 z^4+4 a^3 z^3+18 a z^3+22 z^3 a^{-1} +7 z^3 a^{-3} -z^3 a^{-5} +6 a^6 z^2+9 a^4 z^2-7 a^2 z^2-2 z^2 a^{-2} +2 z^2 a^{-4} -14 z^2+4 a^5 z+2 a^3 z-6 a z-6 z a^{-1} -2 z a^{-3} -4 a^6-6 a^4-a^2+ a^{-2} +3-2 a^5 z^{-1} -2 a^3 z^{-1} +a^6 z^{-2} +2 a^4 z^{-2} +a^2 z^{-2}$ (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                   6 1   -5 5                 9 3     6 3               10 6       -4 1             11 9         2 -1           11 12           1 -3         7 9             -2 -5       4 11               7 -7     3 7                 -4 -9   1 6                   5 -11   1                     -1 -13 1                       1

Integral Khovanov Homology (db, data source)

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

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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