L11a341

L11a340

L11a342

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Link Presentations

[edit Notes on L11a341's Link Presentations]

Planar diagram presentation	X12,1,13,2 X16,8,17,7 X10,5,1,6 X6374 X4,9,5,10 X18,14,19,13 X22,20,11,19 X20,15,21,16 X14,21,15,22 X2,11,3,12 X8,18,9,17
Gauss code	{1, -10, 4, -5, 3, -4, 2, -11, 5, -3}, {10, -1, 6, -9, 8, -2, 11, -6, 7, -8, 9, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(u-1) (v-1) \left(u^2 v^2-2 u^2 v+u^2-2 u v^2+5 u v-2 u+v^2-2 v+1\right)}{u^{3/2} v^{3/2}}$ (db)
Jones polynomial	$-q^{7/2}+4 q^{5/2}-10 q^{3/2}+15 \sqrt{q}-\frac{20}{\sqrt{q}}+\frac{22}{q^{3/2}}-\frac{22}{q^{5/2}}+\frac{18}{q^{7/2}}-\frac{13}{q^{9/2}}+\frac{7}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^7 (-z)-a^7 z^{-1} +3 a^5 z^3+6 a^5 z+4 a^5 z^{-1} -3 a^3 z^5-9 a^3 z^3-12 a^3 z-6 a^3 z^{-1} +a z^7+4 a z^5-z^5 a^{-1} +9 a z^3-2 z^3 a^{-1} +10 a z+5 a z^{-1} -3 z a^{-1} -2 a^{-1} z^{-1}$ (db)
Kauffman polynomial	$-a^4 z^{10}-a^2 z^{10}-3 a^5 z^9-8 a^3 z^9-5 a z^9-4 a^6 z^8-13 a^4 z^8-19 a^2 z^8-10 z^8-3 a^7 z^7-6 a^5 z^7-6 a^3 z^7-12 a z^7-9 z^7 a^{-1} -a^8 z^6+5 a^6 z^6+27 a^4 z^6+38 a^2 z^6-4 z^6 a^{-2} +13 z^6+8 a^7 z^5+29 a^5 z^5+50 a^3 z^5+45 a z^5+15 z^5 a^{-1} -z^5 a^{-3} +3 a^8 z^4+4 a^6 z^4-10 a^4 z^4-19 a^2 z^4+4 z^4 a^{-2} -4 z^4-8 a^7 z^3-34 a^5 z^3-60 a^3 z^3-47 a z^3-12 z^3 a^{-1} +z^3 a^{-3} -3 a^8 z^2-8 a^6 z^2-6 a^4 z^2-z^2 a^{-2} +4 a^7 z+18 a^5 z+31 a^3 z+24 a z+7 z a^{-1} +a^8+3 a^6+3 a^4+a^2+1-a^7 z^{-1} -4 a^5 z^{-1} -6 a^3 z^{-1} -5 a z^{-1} -2 a^{-1} z^{-1}$ (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                     3   -3 4                   7 1   6 2                 8 3     -5 0               12 7       5 -2             12 10         -2 -4           10 10           0 -6         8 12             4 -8       5 10               -5 -10     2 8                 6 -12   1 5                   -4 -14   2                     2 -16 1                       -1

Integral Khovanov Homology (db, data source)

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

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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