L11a41

L11a40

L11a42

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Link Presentations

[edit Notes on L11a41's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(t(1)-1) (t(2)-1)^3 \left(t(2)^2-3 t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}}$ (db)
Jones polynomial	$-\frac{15}{q^{9/2}}-q^{7/2}+\frac{21}{q^{7/2}}+5 q^{5/2}-\frac{26}{q^{5/2}}-12 q^{3/2}+\frac{26}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{3}{q^{13/2}}+\frac{8}{q^{11/2}}+18 \sqrt{q}-\frac{24}{\sqrt{q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^7 (-z)-a^7 z^{-1} +3 a^5 z^3+5 a^5 z+3 a^5 z^{-1} -3 a^3 z^5-7 a^3 z^3-7 a^3 z-3 a^3 z^{-1} +a z^7+3 a z^5-z^5 a^{-1} +5 a z^3-z^3 a^{-1} +4 a z+2 a z^{-1} -z a^{-1} - a^{-1} z^{-1}$ (db)
Kauffman polynomial	$a^8 z^6-3 a^8 z^4+3 a^8 z^2-a^8+3 a^7 z^7-7 a^7 z^5+6 a^7 z^3-3 a^7 z+a^7 z^{-1} +5 a^6 z^8-8 a^6 z^6+2 a^6 z^4+3 a^6 z^2-2 a^6+5 a^5 z^9-a^5 z^7-15 a^5 z^5+21 a^5 z^3-12 a^5 z+3 a^5 z^{-1} +2 a^4 z^{10}+15 a^4 z^8-40 a^4 z^6+32 a^4 z^4-8 a^4 z^2+14 a^3 z^9-12 a^3 z^7-23 a^3 z^5+z^5 a^{-3} +33 a^3 z^3-16 a^3 z+3 a^3 z^{-1} +2 a^2 z^{10}+25 a^2 z^8-58 a^2 z^6+5 z^6 a^{-2} +39 a^2 z^4-3 z^4 a^{-2} -11 a^2 z^2+2 a^2+9 a z^9+4 a z^7+12 z^7 a^{-1} -32 a z^5-16 z^5 a^{-1} +24 a z^3+6 z^3 a^{-1} -9 a z-2 z a^{-1} +2 a z^{-1} + a^{-1} z^{-1} +15 z^8-22 z^6+9 z^4-3 z^2$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     4   -4 4                   8 1   7 2                 10 4     -6 0               14 8       6 -2             14 12         -2 -4           12 12           0 -6         9 14             5 -8       6 12               -6 -10     2 9                 7 -12   1 6                   -5 -14   2                     2 -16 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-3$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=-2$ ${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=-1$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{14}$ ${\mathbb Z}^{14}$ $r=0$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{14}$ $r=1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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