L11a421

L11a420

L11a422

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Link Presentations

[edit Notes on L11a421's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{3 u v^2 w^2-3 u v^2 w+u v^2+3 u v w^3-7 u v w^2+5 u v w-u v+u w^4-4 u w^3+5 u w^2-2 u w+2 v^2 w^3-5 v^2 w^2+4 v^2 w-v^2+v w^4-5 v w^3+7 v w^2-3 v w-w^4+3 w^3-3 w^2}{\sqrt{u} v w^2}$ (db)
Jones polynomial	$q^{-2} -4 q^{-3} +10 q^{-4} -15 q^{-5} +21 q^{-6} -22 q^{-7} +23 q^{-8} -18 q^{-9} +14 q^{-10} -8 q^{-11} +3 q^{-12} - q^{-13}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{14} z^{-2} +4 a^{12} z^{-2} +5 a^{12}-9 z^2 a^{10}-5 a^{10} z^{-2} -13 a^{10}+5 z^4 a^8+9 z^2 a^8+2 a^8 z^{-2} +7 a^8+4 z^4 a^6+5 z^2 a^6+a^6+z^4 a^4$ (db)
Kauffman polynomial	$z^7 a^{15}-4 z^5 a^{15}+6 z^3 a^{15}-4 z a^{15}+a^{15} z^{-1} +3 z^8 a^{14}-10 z^6 a^{14}+12 z^4 a^{14}-7 z^2 a^{14}-a^{14} z^{-2} +3 a^{14}+4 z^9 a^{13}-7 z^7 a^{13}-9 z^5 a^{13}+26 z^3 a^{13}-19 z a^{13}+5 a^{13} z^{-1} +2 z^{10} a^{12}+9 z^8 a^{12}-44 z^6 a^{12}+52 z^4 a^{12}-29 z^2 a^{12}-4 a^{12} z^{-2} +15 a^{12}+13 z^9 a^{11}-23 z^7 a^{11}-17 z^5 a^{11}+48 z^3 a^{11}-33 z a^{11}+9 a^{11} z^{-1} +2 z^{10} a^{10}+22 z^8 a^{10}-71 z^6 a^{10}+69 z^4 a^{10}-42 z^2 a^{10}-5 a^{10} z^{-2} +20 a^{10}+9 z^9 a^9-35 z^5 a^9+35 z^3 a^9-18 z a^9+5 a^9 z^{-1} +16 z^8 a^8-27 z^6 a^8+19 z^4 a^8-15 z^2 a^8-2 a^8 z^{-2} +8 a^8+15 z^7 a^7-19 z^5 a^7+7 z^3 a^7+10 z^6 a^6-9 z^4 a^6+5 z^2 a^6-a^6+4 z^5 a^5+z^4 a^4$ (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 -3                       1 1 -5                     4 1 -3 -7                   6     6 -9                 9 4     -5 -11               12 6       6 -13             11 10         -1 -15           12 11           1 -17         8 13             5 -19       6 10               -4 -21     2 8                 6 -23   1 6                   -5 -25   2                     2 -27 1                       -1

Integral Khovanov Homology (db, data source)

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

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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