L11a484

L11a483

L11a485

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Link Presentations

[edit Notes on L11a484's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X12,6,13,5 X8493 X16,10,17,9 X22,17,19,18 X20,12,21,11 X10,20,11,19 X18,21,5,22 X2,14,3,13
Gauss code	{1, -11, 5, -3}, {9, -8, 10, -7}, {4, -1, 2, -5, 6, -9, 8, -4, 11, -2, 3, -6, 7, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{(u-1) (v-1) (w-1)^3 \left(w^2+1\right)}{\sqrt{u} \sqrt{v} w^{5/2}}$ (db)
Jones polynomial	$q^7-4 q^6+8 q^5-13 q^4- q^{-4} +18 q^3+4 q^{-3} -20 q^2-8 q^{-2} +21 q+14 q^{-1} -16$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^6 a^{-4} +3 z^4 a^{-4} +2 z^2 a^{-4} -z^8 a^{-2} -5 z^6 a^{-2} -a^2 z^4-9 z^4 a^{-2} -2 a^2 z^2-6 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^6+7 z^4+6 z^2-2 z^{-2}$ (db)
Kauffman polynomial	$z^4 a^{-8} +4 z^5 a^{-7} -2 z^3 a^{-7} +8 z^6 a^{-6} -7 z^4 a^{-6} +z^2 a^{-6} +11 z^7 a^{-5} -13 z^5 a^{-5} +4 z^3 a^{-5} -z a^{-5} +12 z^8 a^{-4} -20 z^6 a^{-4} +12 z^4 a^{-4} -5 z^2 a^{-4} +8 z^9 a^{-3} +a^3 z^7-6 z^7 a^{-3} -3 a^3 z^5-18 z^5 a^{-3} +3 a^3 z^3+20 z^3 a^{-3} -a^3 z-4 z a^{-3} +2 z^{10} a^{-2} +4 a^2 z^8+18 z^8 a^{-2} -14 a^2 z^6-67 z^6 a^{-2} +16 a^2 z^4+66 z^4 a^{-2} -8 a^2 z^2-21 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} +5 a z^9+13 z^9 a^{-1} -12 a z^7-30 z^7 a^{-1} -a z^5+z^5 a^{-1} +12 a z^3+23 z^3 a^{-1} -4 a z-6 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +2 z^{10}+10 z^8-53 z^6+62 z^4-23 z^2+2 z^{-2} +1$ (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 15                       1 1 13                     3   -3 11                   5 1   4 9                 8 3     -5 7               10 5       5 5             10 8         -2 3           11 10           1 1         9 14             5 -1       5 7               -2 -3     3 9                 6 -5   1 5                   -4 -7   3                     3 -9 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=0$ ${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{11}$ $r=1$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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