L11a90

L11a89

L11a91

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Link Presentations

[edit Notes on L11a90's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(t(1)-1) (t(2)-1) \left(t(2)^4-4 t(2)^3+3 t(2)^2-4 t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}}$ (db)
Jones polynomial	$q^{17/2}-4 q^{15/2}+8 q^{13/2}-12 q^{11/2}+15 q^{9/2}-17 q^{7/2}+16 q^{5/2}-13 q^{3/2}+9 \sqrt{q}-\frac{6}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^7 a^{-3} -2 z^5 a^{-1} +4 z^5 a^{-3} -2 z^5 a^{-5} +a z^3-7 z^3 a^{-1} +7 z^3 a^{-3} -5 z^3 a^{-5} +z^3 a^{-7} +3 a z-8 z a^{-1} +7 z a^{-3} -3 z a^{-5} +z a^{-7} +2 a z^{-1} -4 a^{-1} z^{-1} +3 a^{-3} z^{-1} - a^{-5} z^{-1}$ (db)
Kauffman polynomial	$-z^{10} a^{-2} -z^{10} a^{-4} -2 z^9 a^{-1} -6 z^9 a^{-3} -4 z^9 a^{-5} -4 z^8 a^{-2} -10 z^8 a^{-4} -8 z^8 a^{-6} -2 z^8-a z^7+2 z^7 a^{-1} +8 z^7 a^{-3} -5 z^7 a^{-5} -10 z^7 a^{-7} +17 z^6 a^{-2} +25 z^6 a^{-4} +7 z^6 a^{-6} -8 z^6 a^{-8} +7 z^6+5 a z^5+13 z^5 a^{-1} +14 z^5 a^{-3} +23 z^5 a^{-5} +13 z^5 a^{-7} -4 z^5 a^{-9} -9 z^4 a^{-2} -9 z^4 a^{-4} +3 z^4 a^{-6} +8 z^4 a^{-8} -z^4 a^{-10} -6 z^4-9 a z^3-26 z^3 a^{-1} -24 z^3 a^{-3} -15 z^3 a^{-5} -6 z^3 a^{-7} +2 z^3 a^{-9} -7 z^2 a^{-2} -6 z^2 a^{-4} -2 z^2 a^{-6} -2 z^2 a^{-8} -z^2+7 a z+17 z a^{-1} +13 z a^{-3} +4 z a^{-5} +z a^{-7} +3 a^{-2} +3 a^{-4} + a^{-6} +2-2 a z^{-1} -4 a^{-1} z^{-1} -3 a^{-3} z^{-1} - a^{-5} z^{-1}$ (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 18                       1 -1 16                     3   3 14                   5 1   -4 12                 7 3     4 10               8 5       -3 8             9 7         2 6           7 8           1 4         6 9             -3 2       5 9               4 0     1 4                 -3 -2   1 5                   4 -4   1                     -1 -6 1                       1

Integral Khovanov Homology (db, data source)

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

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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