L11a32

L11a31

L11a33

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Link Presentations

[edit Notes on L11a32's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{2 t(1) t(2)^3-6 t(2)^3-9 t(1) t(2)^2+12 t(2)^2+12 t(1) t(2)-9 t(2)-6 t(1)+2}{\sqrt{t(1)} t(2)^{3/2}}$ (db)
Jones polynomial	$q^{15/2}-3 q^{13/2}+7 q^{11/2}-11 q^{9/2}+16 q^{7/2}-19 q^{5/2}+18 q^{3/2}-17 \sqrt{q}+\frac{12}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{1}{q^{7/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^5 a^{-1} +z^5 a^{-3} -2 a z^3-2 z^3 a^{-1} -z^3 a^{-3} -2 z^3 a^{-5} +a^3 z+a z-4 z a^{-1} -z a^{-3} +z a^{-7} +2 a z^{-1} -2 a^{-1} z^{-1} - a^{-3} z^{-1} + a^{-5} z^{-1}$ (db)
Kauffman polynomial	$z^6 a^{-8} -3 z^4 a^{-8} +2 z^2 a^{-8} +3 z^7 a^{-7} -8 z^5 a^{-7} +5 z^3 a^{-7} -z a^{-7} +5 z^8 a^{-6} -14 z^6 a^{-6} +15 z^4 a^{-6} -11 z^2 a^{-6} +4 a^{-6} +4 z^9 a^{-5} -5 z^7 a^{-5} -4 z^5 a^{-5} +6 z^3 a^{-5} -z a^{-5} - a^{-5} z^{-1} +z^{10} a^{-4} +12 z^8 a^{-4} -42 z^6 a^{-4} +56 z^4 a^{-4} -36 z^2 a^{-4} +9 a^{-4} +8 z^9 a^{-3} -11 z^7 a^{-3} +a^3 z^5-z^5 a^{-3} -2 a^3 z^3+7 z^3 a^{-3} +a^3 z-z a^{-3} - a^{-3} z^{-1} +z^{10} a^{-2} +13 z^8 a^{-2} +3 a^2 z^6-35 z^6 a^{-2} -4 a^2 z^4+38 z^4 a^{-2} +a^2 z^2-19 z^2 a^{-2} +4 a^{-2} +4 z^9 a^{-1} +6 a z^7+3 z^7 a^{-1} -11 a z^5-17 z^5 a^{-1} +11 a z^3+19 z^3 a^{-1} -8 a z-10 z a^{-1} +2 a z^{-1} +2 a^{-1} z^{-1} +6 z^8-5 z^6-4 z^4+5 z^2-2$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 16                       1 -1 14                     2   2 12                   5 1   -4 10                 6 2     4 8               10 5       -5 6             9 6         3 4           9 10           1 2         8 9             -1 0       5 10               5 -2     3 7                 -4 -4     5                   5 -6 1 3                     -2 -8 1                       1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=2$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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