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L10a117 L10a119

Knotscape

This page is passe. Go here instead! The 2-Component Link L10a118 Visit L10a118's page at Knotilus! Acknowledgement

DrawMorseLink

Further views:

Rich Schwartz' "25"

PD Presentation:

X12,1,13,2 X2,13,3,14 X14,3,15,4 X16,5,17,6 X18,7,19,8 X20,9,11,10 X10,11,1,12 X4,15,5,16 X6,17,7,18 X8,19,9,20

Gauss Code:

{{1, -2, 3, -8, 4, -9, 5, -10, 6, -7}, {7, -1, 2, -3, 8, -4, 9, -5, 10, -6}}

Jones Polynomial:

- q-29/2 + q-27/2 - q-25/2 + q-23/2 - q-21/2 + q-19/2 - q-17/2 + q-15/2 - q-13/2 - q-9/2

A2 (sl(3)) Invariant:

q-42 + q-40 + q-38 + q-24 + q-22 + 2q-20 + q-18 + q-16

HOMFLY-PT Polynomial:

- a9z-1 - 15a9z - 35a9z3 - 28a9z5 - 9a9z7 - a9z9 + a11z-1 + 10a11z + 15a11z3 + 7a11z5 + a11z7

Kauffman Polynomial:

a9z-1 - 15a9z + 35a9z3 - 28a9z5 + 9a9z7 - a9z9 - a10 + 10a10z2 - 15a10z4 + 7a10z6 - a10z8 + a11z-1 - 11a11z + 25a11z3 - 22a11z5 + 8a11z7 - a11z9 + 4a12z2 - 10a12z4 + 6a12z6 - a12z8 + a13z - 6a13z3 + 5a13z5 - a13z7 - 3a14z2 + 4a14z4 - a14z6 - a15z + 3a15z3 - a15z5 + 2a16z2 - a16z4 + a17z - a17z3 - a18z2 - a19z

Khovanov Homology:

trqj r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -8                     1 j = -10                     1 j = -12                 1     j = -14                       j = -16             1 1       j = -18                       j = -20         1 1           j = -22                       j = -24     1 1               j = -26                       j = -28 1 1                   j = -30 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	2
In[3]:=	Show[DrawMorseLink[Link[10, Alternating, 118]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 118]]
Out[4]=	PD[X[12, 1, 13, 2], X[2, 13, 3, 14], X[14, 3, 15, 4], X[16, 5, 17, 6], > X[18, 7, 19, 8], X[20, 9, 11, 10], X[10, 11, 1, 12], X[4, 15, 5, 16], > X[6, 17, 7, 18], X[8, 19, 9, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -8, 4, -9, 5, -10, 6, -7}, > {7, -1, 2, -3, 8, -4, 9, -5, 10, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(29/2) -(27/2) -(25/2) -(23/2) -(21/2) -(19/2) -(17/2) -q + q - q + q - q + q - q + -(15/2) -(13/2) -(9/2) > q - q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-42 -40 -38 -24 -22 2 -18 -16 q + q + q + q + q + --- + q + q 20 q
In[8]:=	HOMFLYPT[Link[10, Alternating, 118]][a, z]
Out[8]=	9 11 a a 9 11 9 3 11 3 9 5 11 5 -(--) + --- - 15 a z + 10 a z - 35 a z + 15 a z - 28 a z + 7 a z - z z 9 7 11 7 9 9 > 9 a z + a z - a z
In[9]:=	Kauffman[Link[10, Alternating, 118]][a, z]
Out[9]=	9 11 10 a a 9 11 13 15 17 19 -a + -- + --- - 15 a z - 11 a z + a z - a z + a z - a z + z z 10 2 12 2 14 2 16 2 18 2 9 3 > 10 a z + 4 a z - 3 a z + 2 a z - a z + 35 a z + 11 3 13 3 15 3 17 3 10 4 12 4 > 25 a z - 6 a z + 3 a z - a z - 15 a z - 10 a z + 14 4 16 4 9 5 11 5 13 5 15 5 10 6 > 4 a z - a z - 28 a z - 22 a z + 5 a z - a z + 7 a z + 12 6 14 6 9 7 11 7 13 7 10 8 12 8 9 9 > 6 a z - a z + 9 a z + 8 a z - a z - a z - a z - a z - 11 9 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	-10 -8 1 1 1 1 1 1 1 q + q + ------- + ------- + ------ + ------ + ------ + ------ + ------ + 30 10 28 10 28 9 24 8 24 7 20 6 20 5 q t q t q t q t q t q t q t 1 1 1 > ------ + ------ + ------ 16 4 16 3 12 2 q t q t q t

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