L11n337

L11n336

L11n338

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Link Presentations

[edit Notes on L11n337's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,18,12,19 X7,14,8,15 X13,8,14,9 X19,22,20,13 X15,20,16,21 X21,16,22,17 X17,12,18,5 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, -3, 9}, {-5, 4, -7, 8, -9, 3, -6, 7, -8, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(t(2)-1) \left(t(1) t(3)^3+t(1) t(2) t(3)^3+2 t(2)^2 t(3)^2-2 t(1) t(3)^2-2 t(2)^2 t(3)+2 t(1) t(3)+t(2)^2+t(2)\right)}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}}$ (db)
Jones polynomial	$- q^{-13} +2 q^{-12} -4 q^{-11} +6 q^{-10} -5 q^{-9} +6 q^{-8} -4 q^{-7} +4 q^{-6} - q^{-5} + q^{-3}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{14} z^{-2} +4 a^{12} z^{-2} +4 a^{12}-5 a^{10} z^2-5 a^{10} z^{-2} -9 a^{10}+a^8 z^4+a^8 z^2+2 a^8 z^{-2} +2 a^8+a^6 z^6+6 a^6 z^4+8 a^6 z^2+3 a^6$ (db)
Kauffman polynomial	$a^{15} z^7-5 a^{15} z^5+8 a^{15} z^3-5 a^{15} z+a^{15} z^{-1} +2 a^{14} z^8-9 a^{14} z^6+12 a^{14} z^4-8 a^{14} z^2-a^{14} z^{-2} +4 a^{14}+a^{13} z^9+a^{13} z^7-21 a^{13} z^5+36 a^{13} z^3-23 a^{13} z+5 a^{13} z^{-1} +6 a^{12} z^8-28 a^{12} z^6+41 a^{12} z^4-34 a^{12} z^2-4 a^{12} z^{-2} +18 a^{12}+a^{11} z^9+3 a^{11} z^7-32 a^{11} z^5+57 a^{11} z^3-39 a^{11} z+9 a^{11} z^{-1} +4 a^{10} z^8-21 a^{10} z^6+38 a^{10} z^4-37 a^{10} z^2-5 a^{10} z^{-2} +21 a^{10}+3 a^9 z^7-17 a^9 z^5+30 a^9 z^3-20 a^9 z+5 a^9 z^{-1} -a^8 z^6+3 a^8 z^4-3 a^8 z^2-2 a^8 z^{-2} +5 a^8-a^7 z^5+a^7 z^3+a^7 z+a^6 z^6-6 a^6 z^4+8 a^6 z^2-3 a^6$ (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -5                       1 1 -7                       1 1 -9                 2 1     -1 -11               3         3 -13             4 5 1       0 -15           3 1           2 -17         2 4 1           1 -19       4 3               1 -21     1 3                 2 -23   1 3                   -2 -25   1                     1 -27 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-7$ $i=-5$ $i=-3$ $r=-11$ ${\mathbb Z}$ $r=-10$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=-7$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-5$ ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-4$ ${\mathbb Z}_2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

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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