L11a81

L11a80

L11a82

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Link Presentations

[edit Notes on L11a81's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{5 u v^4-11 u v^3+12 u v^2-7 u v+2 u+2 v^5-7 v^4+12 v^3-11 v^2+5 v}{\sqrt{u} v^{5/2}}$ (db)
Jones polynomial	$-\frac{11}{q^{9/2}}+\frac{4}{q^{7/2}}-\frac{1}{q^{5/2}}+\frac{1}{q^{27/2}}-\frac{3}{q^{25/2}}+\frac{8}{q^{23/2}}-\frac{14}{q^{21/2}}+\frac{20}{q^{19/2}}-\frac{24}{q^{17/2}}+\frac{24}{q^{15/2}}-\frac{22}{q^{13/2}}+\frac{16}{q^{11/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{13} (-z)-a^{13} z^{-1} +3 a^{11} z^3+4 a^{11} z+a^{11} z^{-1} -2 a^9 z^5-a^9 z^3+4 a^9 z+2 a^9 z^{-1} -4 a^7 z^5-11 a^7 z^3-9 a^7 z-2 a^7 z^{-1} -a^5 z^5-a^5 z^3$ (db)
Kauffman polynomial	$-z^6 a^{16}+3 z^4 a^{16}-3 z^2 a^{16}+a^{16}-3 z^7 a^{15}+7 z^5 a^{15}-5 z^3 a^{15}+z a^{15}-5 z^8 a^{14}+9 z^6 a^{14}-4 z^4 a^{14}-5 z^9 a^{13}+4 z^7 a^{13}+5 z^5 a^{13}-4 z^3 a^{13}-z a^{13}+a^{13} z^{-1} -2 z^{10} a^{12}-12 z^8 a^{12}+34 z^6 a^{12}-31 z^4 a^{12}+15 z^2 a^{12}-3 a^{12}-13 z^9 a^{11}+18 z^7 a^{11}-3 z^5 a^{11}+2 z^3 a^{11}-4 z a^{11}+a^{11} z^{-1} -2 z^{10} a^{10}-19 z^8 a^{10}+45 z^6 a^{10}-31 z^4 a^{10}+7 z^2 a^{10}-8 z^9 a^9+z^7 a^9+17 z^5 a^9-15 z^3 a^9+7 z a^9-2 a^9 z^{-1} -12 z^8 a^8+17 z^6 a^8-4 z^4 a^8-5 z^2 a^8+3 a^8-10 z^7 a^7+17 z^5 a^7-15 z^3 a^7+9 z a^7-2 a^7 z^{-1} -4 z^6 a^6+3 z^4 a^6-z^5 a^5+z^3 a^5$ (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 -4                       1 1 -6                     4 1 -3 -8                   7     7 -10                 9 4     -5 -12               13 7       6 -14             12 10         -2 -16           12 12           0 -18         8 12             4 -20       6 12               -6 -22     2 8                 6 -24   1 6                   -5 -26   2                     2 -28 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-11$ ${\mathbb Z}$ $r=-10$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-8$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-7$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-6$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=-5$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=-4$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{13}$ $r=-3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-1$ ${\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

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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