L11n376

L11n375

L11n377

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Link Presentations

[edit Notes on L11n376's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X5,10,6,11 X3849 X22,18,19,17 X11,20,12,21 X19,12,20,13 X18,22,5,21 X9,16,10,17 X13,2,14,3
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 $u$, $v$, $w$, ...)	$-\frac{(v-1) (w-1) \left(u w^2-1\right)}{\sqrt{u} \sqrt{v} w^{3/2}}$ (db)
Jones polynomial	$- q^{-9} + q^{-8} -2 q^{-7} +4 q^{-6} -2 q^{-5} +3 q^{-4} - q^{-3} +2 q^{-2}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{10} z^{-2} -a^{10}+a^8 z^4+5 a^8 z^2+4 a^8 z^{-2} +7 a^8-a^6 z^6-6 a^6 z^4-13 a^6 z^2-5 a^6 z^{-2} -13 a^6+2 a^4 z^4+8 a^4 z^2+2 a^4 z^{-2} +7 a^4$ (db)
Kauffman polynomial	$z^3 a^{11}-3 z a^{11}+a^{11} z^{-1} +z^4 a^{10}-2 z^2 a^{10}-a^{10} z^{-2} +2 a^{10}+z^7 a^9-6 z^5 a^9+15 z^3 a^9-14 z a^9+5 a^9 z^{-1} +z^8 a^8-6 z^6 a^8+14 z^4 a^8-13 z^2 a^8-4 a^8 z^{-2} +11 a^8+2 z^7 a^7-11 z^5 a^7+25 z^3 a^7-24 z a^7+9 a^7 z^{-1} +z^8 a^6-6 z^6 a^6+16 z^4 a^6-22 z^2 a^6-5 a^6 z^{-2} +16 a^6+z^7 a^5-5 z^5 a^5+11 z^3 a^5-13 z a^5+5 a^5 z^{-1} +3 z^4 a^4-11 z^2 a^4-2 a^4 z^{-2} +8 a^4$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 χ -3               2 2 -5             1 2 1 -7           2     2 -9       1 1 1     1 -11       4 2       2 -13       2         2 -15   1 2           -1 -17                 0 -19 1               -1

Integral Khovanov Homology (db, data source)

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

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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