L11a427

L11a426

L11a428

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Link Presentations

[edit Notes on L11a427's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X14,5,15,6 X20,7,21,8 X8,19,9,20 X16,12,17,11 X10,13,5,14 X22,18,11,17 X18,22,19,21 X2,9,3,10 X4,16,1,15
Gauss code	{1, -10, 2, -11}, {3, -1, 4, -5, 10, -7}, {6, -2, 7, -3, 11, -6, 8, -9, 5, -4, 9, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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