L11a406

L11a405

L11a407

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Link Presentations

[edit Notes on L11a406's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{2 u v^2 w-2 u v^2+2 u v w^2-7 u v w+4 u v-2 u w^2+4 u w-2 u+2 v^2 w^2-4 v^2 w+2 v^2-4 v w^2+7 v w-2 v+2 w^2-2 w}{\sqrt{u} v w}$ (db)
Jones polynomial	$- q^{-10} +3 q^{-9} -6 q^{-8} +10 q^{-7} -13 q^{-6} +16 q^{-5} -15 q^{-4} +15 q^{-3} -10 q^{-2} +q+7 q^{-1} -3$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{10}+3 a^8 z^2+2 a^8-2 a^6 z^4-a^6 z^2+a^6 z^{-2} -3 a^4 z^4-4 a^4 z^2-2 a^4 z^{-2} -4 a^4-a^2 z^4+2 a^2 z^2+a^2 z^{-2} +3 a^2+z^2$ (db)
Kauffman polynomial	$z^7 a^{11}-4 z^5 a^{11}+5 z^3 a^{11}-2 z a^{11}+3 z^8 a^{10}-12 z^6 a^{10}+15 z^4 a^{10}-7 z^2 a^{10}+2 a^{10}+3 z^9 a^9-7 z^7 a^9-3 z^5 a^9+12 z^3 a^9-4 z a^9+z^{10} a^8+7 z^8 a^8-34 z^6 a^8+42 z^4 a^8-21 z^2 a^8+4 a^8+7 z^9 a^7-15 z^7 a^7+3 z^5 a^7+2 z^3 a^7+z^{10} a^6+10 z^8 a^6-30 z^6 a^6+23 z^4 a^6-7 z^2 a^6+a^6 z^{-2} -2 a^6+4 z^9 a^5-8 z^5 a^5-2 z^3 a^5+6 z a^5-2 a^5 z^{-1} +6 z^8 a^4-2 z^6 a^4-13 z^4 a^4+16 z^2 a^4+2 a^4 z^{-2} -7 a^4+7 z^7 a^3-7 z^5 a^3+z^3 a^3+4 z a^3-2 a^3 z^{-1} +6 z^6 a^2-8 z^4 a^2+8 z^2 a^2+a^2 z^{-2} -4 a^2+3 z^5 a-2 z^3 a+z^4-z^2$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 3                       1 1 1                     2   -2 -1                   5 1   4 -3                 6 3     -3 -5               9 4       5 -7             8 8         0 -9           8 7           1 -11         6 9             3 -13       4 7               -3 -15     2 6                 4 -17   1 4                   -3 -19   2                     2 -21 1                       -1

Integral Khovanov Homology (db, data source)

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

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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