L10a143

L10a142

L10a144

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Link Presentations

[edit Notes on L10a143's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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