L11n377

L11n376

L11n378

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Link Presentations

[edit Notes on L11n377's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X15,1,16,4 X5,10,6,11 X3849 X17,22,18,19 X11,20,12,21 X19,12,20,13 X21,18,22,5 X9,16,10,17 X2,14,3,13
Gauss code	{1, -11, -5, 3}, {-8, 7, -9, 6}, {-4, -1, 2, 5, -10, 4, -7, 8, 11, -2, -3, 10, -6, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{(t(1)-1) (t(3)-1) \left(t(3)^2+2 t(2) t(3)-2 t(3)-t(2)\right)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}}$ (db)
Jones polynomial	$- q^{-10} +2 q^{-9} -4 q^{-8} +6 q^{-7} -7 q^{-6} +9 q^{-5} -6 q^{-4} +7 q^{-3} -4 q^{-2} +2 q^{-1}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{10} z^{-2} -a^{10}+3 a^8 z^2+4 a^8 z^{-2} +6 a^8-2 a^6 z^4-6 a^6 z^2-5 a^6 z^{-2} -10 a^6-a^4 z^4+a^4 z^2+2 a^4 z^{-2} +4 a^4+2 a^2 z^2+a^2$ (db)
Kauffman polynomial	$z^7 a^{11}-5 z^5 a^{11}+8 z^3 a^{11}-5 z a^{11}+a^{11} z^{-1} +2 z^8 a^{10}-9 z^6 a^{10}+12 z^4 a^{10}-6 z^2 a^{10}-a^{10} z^{-2} +3 a^{10}+z^9 a^9+z^7 a^9-19 z^5 a^9+33 z^3 a^9-20 z a^9+5 a^9 z^{-1} +6 z^8 a^8-24 z^6 a^8+31 z^4 a^8-23 z^2 a^8-4 a^8 z^{-2} +13 a^8+z^9 a^7+5 z^7 a^7-29 z^5 a^7+41 z^3 a^7-30 z a^7+9 a^7 z^{-1} +4 z^8 a^6-12 z^6 a^6+18 z^4 a^6-25 z^2 a^6-5 a^6 z^{-2} +16 a^6+5 z^7 a^5-14 z^5 a^5+19 z^3 a^5-15 z a^5+5 a^5 z^{-1} +3 z^6 a^4-z^4 a^4-5 z^2 a^4-2 a^4 z^{-2} +6 a^4+z^5 a^3+3 z^3 a^3+3 z^2 a^2-a^2$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -1                   2 2 -3                 4 2 -2 -5               3     3 -7             3 4     1 -9           6 3       3 -11         3 5         2 -13       3 4           -1 -15     1 3             2 -17   1 3               -2 -19   1                 1 -21 1                   -1

Integral Khovanov Homology (db, data source)

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

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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