L11a207

L11a206

L11a208

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Link Presentations

[edit Notes on L11a207's Link Presentations]

Planar diagram presentation	X8192 X14,9,15,10 X6718 X22,15,7,16 X16,6,17,5 X4,22,5,21 X10,4,11,3 X20,18,21,17 X12,20,13,19 X18,12,19,11 X2,14,3,13
Gauss code	{1, -11, 7, -6, 5, -3}, {3, -1, 2, -7, 10, -9, 11, -2, 4, -5, 8, -10, 9, -8, 6, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{\left(u v^2-2 u v+u-v^2+2 v-2\right) \left(2 u v^2-2 u v+u-v^2+2 v-1\right)}{u v^2}$ (db)
Jones polynomial	$q^{15/2}-5 q^{13/2}+11 q^{11/2}-17 q^{9/2}+23 q^{7/2}-27 q^{5/2}+25 q^{3/2}-23 \sqrt{q}+\frac{16}{\sqrt{q}}-\frac{9}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{1}{q^{7/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^5 a^{-5} +z^3 a^{-5} -z a^{-5} -z^7 a^{-3} -2 z^5 a^{-3} +z^3 a^{-3} +3 z a^{-3} - a^{-3} z^{-1} -z^7 a^{-1} +a z^5-3 z^5 a^{-1} +2 a z^3-4 z^3 a^{-1} +a z-2 z a^{-1} + a^{-1} z^{-1}$ (db)
Kauffman polynomial	$z^6 a^{-8} -z^4 a^{-8} +5 z^7 a^{-7} -9 z^5 a^{-7} +3 z^3 a^{-7} +10 z^8 a^{-6} -22 z^6 a^{-6} +12 z^4 a^{-6} +9 z^9 a^{-5} -12 z^7 a^{-5} -5 z^5 a^{-5} +7 z^3 a^{-5} -2 z a^{-5} +3 z^{10} a^{-4} +16 z^8 a^{-4} -47 z^6 a^{-4} +29 z^4 a^{-4} -2 z^2 a^{-4} +17 z^9 a^{-3} -26 z^7 a^{-3} +a^3 z^5-a^3 z^3+12 z^3 a^{-3} -5 z a^{-3} - a^{-3} z^{-1} +3 z^{10} a^{-2} +16 z^8 a^{-2} +4 a^2 z^6-39 z^6 a^{-2} -5 a^2 z^4+23 z^4 a^{-2} +2 a^2 z^2-2 z^2 a^{-2} + a^{-2} +8 z^9 a^{-1} +8 a z^7-z^7 a^{-1} -11 a z^5-16 z^5 a^{-1} +7 a z^3+16 z^3 a^{-1} -2 a z-5 z a^{-1} - a^{-1} z^{-1} +10 z^8-11 z^6+2 z^4+2 z^2$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 16                       1 -1 14                     4   4 12                   7 1   -6 10                 10 4     6 8               13 7       -6 6             14 10         4 4           12 14           2 2         11 13             -2 0       6 13               7 -2     3 10                 -7 -4   1 6                   5 -6   3                     -3 -8 1                       1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{11}$ $r=1$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=2$ ${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{14}$ $r=3$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{13}$ $r=4$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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