L11a492

L11a491

L11a493

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Link Presentations

[edit Notes on L11a492's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(u-1) (v-1) (w-1) \left(w^2-w+1\right)^2}{\sqrt{u} \sqrt{v} w^{5/2}}$ (db)
Jones polynomial	$q^6-4 q^5+9 q^4-15 q^3+20 q^2-22 q+25-19 q^{-1} +15 q^{-2} -9 q^{-3} +4 q^{-4} - q^{-5}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^4 a^{-4} +2 z^2 a^{-4} + a^{-4} -a^2 z^6-2 z^6 a^{-2} -3 a^2 z^4-7 z^4 a^{-2} -3 a^2 z^2-8 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -a^2-3 a^{-2} +z^8+5 z^6+10 z^4+9 z^2-2 z^{-2} +3$ (db)
Kauffman polynomial	$2 z^{10} a^{-2} +2 z^{10}+8 a z^9+14 z^9 a^{-1} +6 z^9 a^{-3} +11 a^2 z^8+16 z^8 a^{-2} +7 z^8 a^{-4} +20 z^8+8 a^3 z^7-8 a z^7-24 z^7 a^{-1} -4 z^7 a^{-3} +4 z^7 a^{-5} +4 a^4 z^6-23 a^2 z^6-55 z^6 a^{-2} -15 z^6 a^{-4} +z^6 a^{-6} -66 z^6+a^5 z^5-12 a^3 z^5-5 a z^5-17 z^5 a^{-3} -9 z^5 a^{-5} -5 a^4 z^4+24 a^2 z^4+60 z^4 a^{-2} +9 z^4 a^{-4} -2 z^4 a^{-6} +78 z^4-a^5 z^3+4 a^3 z^3+13 a z^3+21 z^3 a^{-1} +19 z^3 a^{-3} +6 z^3 a^{-5} -13 a^2 z^2-30 z^2 a^{-2} -6 z^2 a^{-4} +z^2 a^{-6} -36 z^2-a^3 z-5 a z-9 z a^{-1} -7 z a^{-3} -2 z a^{-5} +2 a^2+6 a^{-2} +2 a^{-4} +7-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 13                       1 1 11                     3   -3 9                   6 1   5 7                 9 3     -6 5               11 6       5 3             11 9         -2 1           14 11           3 -1         9 15             6 -3       6 10               -4 -5     3 9                 6 -7   1 6                   -5 -9   3                     3 -11 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=0$ ${\mathbb Z}^{15}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{14}$ $r=1$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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