L11a400

L11a399

L11a401

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Link Presentations

[edit Notes on L11a400's Link Presentations]

Planar diagram presentation	X6172 X12,6,13,5 X8493 X2,14,3,13 X14,7,15,8 X18,12,19,11 X20,9,21,10 X10,19,5,20 X4,15,1,16 X22,18,11,17 X16,22,17,21
Gauss code	{1, -4, 3, -9}, {2, -1, 5, -3, 7, -8}, {6, -2, 4, -5, 9, -11, 10, -6, 8, -7, 11, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{(u-1) (v-1)^2 (w-1)^2 \left(w^2+1\right)}{\sqrt{u} v w^2}$ (db)
Jones polynomial	$q^7-4 q^6+8 q^5-14 q^4+18 q^3-20 q^2+21 q-16+14 q^{-1} -7 q^{-2} +4 q^{-3} - q^{-4}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^8 a^{-2} -5 z^6 a^{-2} +z^6 a^{-4} +2 z^6-a^2 z^4-9 z^4 a^{-2} +3 z^4 a^{-4} +7 z^4-2 a^2 z^2-5 z^2 a^{-2} +2 z^2 a^{-4} +5 z^2+a^2+4 a^{-2} - a^{-4} -4+2 a^2 z^{-2} +4 a^{-2} z^{-2} - a^{-4} z^{-2} -5 z^{-2}$ (db)
Kauffman polynomial	$2 z^{10} a^{-2} +2 z^{10}+5 a z^9+13 z^9 a^{-1} +8 z^9 a^{-3} +4 a^2 z^8+18 z^8 a^{-2} +13 z^8 a^{-4} +9 z^8+a^3 z^7-14 a z^7-32 z^7 a^{-1} -5 z^7 a^{-3} +12 z^7 a^{-5} -15 a^2 z^6-67 z^6 a^{-2} -24 z^6 a^{-4} +8 z^6 a^{-6} -50 z^6-3 a^3 z^5+7 a z^5+11 z^5 a^{-1} -19 z^5 a^{-3} -16 z^5 a^{-5} +4 z^5 a^{-7} +18 a^2 z^4+68 z^4 a^{-2} +18 z^4 a^{-4} -6 z^4 a^{-6} +z^4 a^{-8} +61 z^4+2 a^3 z^3+a z^3+6 z^3 a^{-1} +15 z^3 a^{-3} +6 z^3 a^{-5} -2 z^3 a^{-7} -7 a^2 z^2-24 z^2 a^{-2} -8 z^2 a^{-4} -23 z^2+5 a z+9 z a^{-1} +5 z a^{-3} +z a^{-5} -3 a^2-2 a^{-2} -4-5 a z^{-1} -9 a^{-1} z^{-1} -5 a^{-3} z^{-1} - a^{-5} z^{-1} +2 a^2 z^{-2} +4 a^{-2} z^{-2} + a^{-4} z^{-2} +5 z^{-2}$ (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 15                       1 1 13                     3   -3 11                   5 1   4 9                 9 3     -6 7               9 5       4 5             11 9         -2 3           10 9           1 1         8 13             5 -1       6 8               -2 -3     3 10                 7 -5   1 4                   -3 -7   3                     3 -9 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=0$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{10}$ $r=1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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