L11a79

L11a78

L11a80

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Link Presentations

[edit Notes on L11a79's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X16,8,17,7 X22,18,5,17 X18,14,19,13 X14,22,15,21 X20,10,21,9 X8,16,9,15 X10,20,11,19 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -8, 7, -9, 11, -2, 5, -6, 8, -3, 4, -5, 9, -7, 6, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{3 t(1) t(2)^3-7 t(2)^3-11 t(1) t(2)^2+14 t(2)^2+14 t(1) t(2)-11 t(2)-7 t(1)+3}{\sqrt{t(1)} t(2)^{3/2}}$ (db)
Jones polynomial	$-15 q^{9/2}+20 q^{7/2}-\frac{1}{q^{7/2}}-23 q^{5/2}+\frac{3}{q^{5/2}}+22 q^{3/2}-\frac{8}{q^{3/2}}+q^{15/2}-4 q^{13/2}+10 q^{11/2}-20 \sqrt{q}+\frac{13}{\sqrt{q}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z a^{-7} -3 z^3 a^{-5} -z a^{-5} + a^{-5} z^{-1} +2 z^5 a^{-3} +2 z^3 a^{-3} +a^3 z+z a^{-3} - a^{-3} z^{-1} +z^5 a^{-1} -2 a z^3-3 z^3 a^{-1} +a z-5 z a^{-1} +2 a z^{-1} -2 a^{-1} z^{-1}$ (db)
Kauffman polynomial	$z^6 a^{-8} -2 z^4 a^{-8} +z^2 a^{-8} +4 z^7 a^{-7} -8 z^5 a^{-7} +4 z^3 a^{-7} -z a^{-7} +8 z^8 a^{-6} -19 z^6 a^{-6} +15 z^4 a^{-6} -10 z^2 a^{-6} +4 a^{-6} +7 z^9 a^{-5} -9 z^7 a^{-5} -7 z^5 a^{-5} +9 z^3 a^{-5} -2 z a^{-5} - a^{-5} z^{-1} +2 z^{10} a^{-4} +16 z^8 a^{-4} -54 z^6 a^{-4} +59 z^4 a^{-4} -34 z^2 a^{-4} +9 a^{-4} +13 z^9 a^{-3} -21 z^7 a^{-3} +a^3 z^5+2 z^5 a^{-3} -2 a^3 z^3+9 z^3 a^{-3} +a^3 z-2 z a^{-3} - a^{-3} z^{-1} +2 z^{10} a^{-2} +15 z^8 a^{-2} +3 a^2 z^6-44 z^6 a^{-2} -4 a^2 z^4+46 z^4 a^{-2} +a^2 z^2-20 z^2 a^{-2} +4 a^{-2} +6 z^9 a^{-1} +6 a z^7-2 z^7 a^{-1} -10 a z^5-10 z^5 a^{-1} +10 a z^3+16 z^3 a^{-1} -7 a z-9 z a^{-1} +2 a z^{-1} +2 a^{-1} z^{-1} +7 z^8-7 z^6+4 z^2-2$ (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                     3   3 12                   7 1   -6 10                 8 3     5 8               12 7       -5 6             11 8         3 4           11 12           1 2         9 11             -2 0       5 12               7 -2     3 8                 -5 -4     5                   5 -6 1 3                     -2 -8 1                       1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{9}$ $r=1$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=2$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=3$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=4$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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