L11a72

L11a71

L11a73

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Link Presentations

[edit Notes on L11a72's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X20,13,21,14 X16,7,17,8 X18,9,19,10 X8,17,9,18 X10,19,11,20 X22,15,5,16 X14,21,15,22 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -6, 5, -7, 11, -2, 3, -9, 8, -4, 6, -5, 7, -3, 9, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{-2 u v^6+3 u v^5-4 u v^4+4 u v^3-3 u v^2+2 u v-u-v^7+2 v^6-3 v^5+4 v^4-4 v^3+3 v^2-2 v}{\sqrt{u} v^{7/2}}$ (db)
Jones polynomial	$-\frac{1}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{5}{q^{11/2}}+\frac{7}{q^{13/2}}-\frac{11}{q^{15/2}}+\frac{11}{q^{17/2}}-\frac{12}{q^{19/2}}+\frac{11}{q^{21/2}}-\frac{8}{q^{23/2}}+\frac{5}{q^{25/2}}-\frac{2}{q^{27/2}}+\frac{1}{q^{29/2}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$a^{13} \left(-z^3\right)-4 a^{13} z-3 a^{13} z^{-1} +3 a^{11} z^5+14 a^{11} z^3+19 a^{11} z+7 a^{11} z^{-1} -2 a^9 z^7-11 a^9 z^5-20 a^9 z^3-15 a^9 z-4 a^9 z^{-1} -a^7 z^7-5 a^7 z^5-7 a^7 z^3-2 a^7 z$ (db)
Kauffman polynomial	$a^{18} z^4-2 a^{18} z^2+a^{18}+2 a^{17} z^5-2 a^{17} z^3+3 a^{16} z^6-2 a^{16} z^4+4 a^{15} z^7-5 a^{15} z^5+4 a^{15} z^3+4 a^{14} z^8-6 a^{14} z^6+4 a^{14} z^4+a^{14} z^2+3 a^{13} z^9-5 a^{13} z^7+6 a^{13} z^5-11 a^{13} z^3+11 a^{13} z-3 a^{13} z^{-1} +a^{12} z^{10}+4 a^{12} z^8-20 a^{12} z^6+30 a^{12} z^4-24 a^{12} z^2+7 a^{12}+6 a^{11} z^9-24 a^{11} z^7+44 a^{11} z^5-53 a^{11} z^3+30 a^{11} z-7 a^{11} z^{-1} +a^{10} z^{10}+2 a^{10} z^8-19 a^{10} z^6+31 a^{10} z^4-24 a^{10} z^2+7 a^{10}+3 a^9 z^9-14 a^9 z^7+26 a^9 z^5-29 a^9 z^3+17 a^9 z-4 a^9 z^{-1} +2 a^8 z^8-8 a^8 z^6+8 a^8 z^4-a^8 z^2+a^7 z^7-5 a^7 z^5+7 a^7 z^3-2 a^7 z$ (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -6                       1 1 -8                     2 1 -1 -10                   3     3 -12                 4 2     -2 -14               7 3       4 -16             5 5         0 -18           7 6           1 -20         4 5             1 -22       4 7               -3 -24     1 4                 3 -26   1 4                   -3 -28   1                     1 -30 1                       -1

Integral Khovanov Homology (db, data source)

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

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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