L10a38

L10a37

L10a39

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Link Presentations

[edit Notes on L10a38's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{\left(v^2-v+1\right) \left(3 u v^2-3 u v+u+v^3-3 v^2+3 v\right)}{\sqrt{u} v^{5/2}}$ (db)
Jones polynomial	$-\frac{8}{q^{9/2}}+\frac{3}{q^{7/2}}-\frac{1}{q^{5/2}}-\frac{1}{q^{25/2}}+\frac{3}{q^{23/2}}-\frac{6}{q^{21/2}}+\frac{11}{q^{19/2}}-\frac{13}{q^{17/2}}+\frac{14}{q^{15/2}}-\frac{14}{q^{13/2}}+\frac{10}{q^{11/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{13} z^{-1} -4 z a^{11}-5 a^{11} z^{-1} +6 z^3 a^9+14 z a^9+8 a^9 z^{-1} -3 z^5 a^7-10 z^3 a^7-11 z a^7-4 a^7 z^{-1} -z^5 a^5-2 z^3 a^5-z a^5$ (db)
Kauffman polynomial	$a^{15} z^5-2 a^{15} z^3+a^{15} z+3 a^{14} z^6-6 a^{14} z^4+5 a^{14} z^2-2 a^{14}+4 a^{13} z^7-4 a^{13} z^5-a^{13} z^3+a^{13} z+a^{13} z^{-1} +3 a^{12} z^8+5 a^{12} z^6-21 a^{12} z^4+22 a^{12} z^2-9 a^{12}+a^{11} z^9+11 a^{11} z^7-23 a^{11} z^5+16 a^{11} z^3-9 a^{11} z+5 a^{11} z^{-1} +7 a^{10} z^8+a^{10} z^6-29 a^{10} z^4+35 a^{10} z^2-14 a^{10}+a^9 z^9+13 a^9 z^7-32 a^9 z^5+32 a^9 z^3-22 a^9 z+8 a^9 z^{-1} +4 a^8 z^8+2 a^8 z^6-18 a^8 z^4+19 a^8 z^2-8 a^8+6 a^7 z^7-13 a^7 z^5+15 a^7 z^3-12 a^7 z+4 a^7 z^{-1} +3 a^6 z^6-4 a^6 z^4+a^6 z^2+a^5 z^5-2 a^5 z^3+a^5 z$ (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 -4                     1 1 -6                   3 1 -2 -8                 5     5 -10               5 3     -2 -12             9 5       4 -14           6 6         0 -16         7 8           -1 -18       4 6             2 -20     2 7               -5 -22   1 4                 3 -24   2                   -2 -26 1                     1

Integral Khovanov Homology (db, data source)

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

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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