L11a483

L11a482

L11a484

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Link Presentations

[edit Notes on L11a483's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X10,6,11,5 X8493 X22,18,19,17 X20,12,21,11 X12,20,13,19 X18,22,5,21 X16,10,17,9 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 $u$, $v$, $w$, ...)	$\frac{(u-1) (w-1)^2 \left(2 v w^2-2 v w+v+w^3-2 w^2+2 w\right)}{\sqrt{u} \sqrt{v} w^{5/2}}$ (db)
Jones polynomial	$q^9-4 q^8+9 q^7-15 q^6+22 q^5-25 q^4+27 q^3-22 q^2+18 q-11+5 q^{-1} - q^{-2}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^2 a^{-8} + a^{-8} -3 z^4 a^{-6} -6 z^2 a^{-6} + a^{-6} z^{-2} -3 a^{-6} +2 z^6 a^{-4} +6 z^4 a^{-4} +8 z^2 a^{-4} -2 a^{-4} z^{-2} +2 a^{-4} +z^6 a^{-2} -3 z^2 a^{-2} + a^{-2} z^{-2} - a^{-2} -z^4+1$ (db)
Kauffman polynomial	$2 z^{10} a^{-4} +2 z^{10} a^{-6} +9 z^9 a^{-3} +15 z^9 a^{-5} +6 z^9 a^{-7} +14 z^8 a^{-2} +26 z^8 a^{-4} +19 z^8 a^{-6} +7 z^8 a^{-8} +11 z^7 a^{-1} -17 z^7 a^{-5} -2 z^7 a^{-7} +4 z^7 a^{-9} -22 z^6 a^{-2} -70 z^6 a^{-4} -59 z^6 a^{-6} -15 z^6 a^{-8} +z^6 a^{-10} +5 z^6+a z^5-14 z^5 a^{-1} -22 z^5 a^{-3} -18 z^5 a^{-5} -20 z^5 a^{-7} -9 z^5 a^{-9} +14 z^4 a^{-2} +65 z^4 a^{-4} +60 z^4 a^{-6} +11 z^4 a^{-8} -2 z^4 a^{-10} -4 z^4+3 z^3 a^{-1} +15 z^3 a^{-3} +29 z^3 a^{-5} +23 z^3 a^{-7} +6 z^3 a^{-9} -9 z^2 a^{-2} -27 z^2 a^{-4} -25 z^2 a^{-6} -6 z^2 a^{-8} +z^2 a^{-10} -z a^{-1} -z a^{-3} -5 z a^{-5} -7 z a^{-7} -2 z a^{-9} + a^{-2} +2 a^{-4} +3 a^{-6} +2 a^{-8} +1-2 a^{-3} z^{-1} -2 a^{-5} z^{-1} + a^{-2} z^{-2} +2 a^{-4} z^{-2} + a^{-6} z^{-2}$ (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   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 19                       1 1 17                     3   -3 15                   6 1   5 13                 9 3     -6 11               13 6       7 9             14 11         -3 7           13 11           2 5         9 14             5 3       9 13               -4 1     4 11                 7 -1   1 7                   -6 -3   4                     4 -5 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{9}$ $r=1$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=2$ ${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{13}$ $r=3$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{14}$ ${\mathbb Z}^{14}$ $r=4$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{13}$ $r=5$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=7$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=8$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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