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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-27/2 + q-25/2 - 2q-23/2 + 3q-21/2 - 3q-19/2 + 3q-17/2 - 3q-15/2 + 2q-13/2 - 2q-11/2 + q-9/2 - q-7/2

A2 (sl(3)) Invariant:

q-40 + q-38 + q-36 + q-34 - q-28 + q-26 + q-24 + q-22 + q-20 + q-16 + q-12

HOMFLY-PT Polynomial:

- 4a7z - 10a7z3 - 6a7z5 - a7z7 - a9z-1 - 7a9z - 11a9z3 - 6a9z5 - a9z7 + a11z-1 + 6a11z + 5a11z3 + a11z5

Kauffman Polynomial:

4a7z - 10a7z3 + 6a7z5 - a7z7 + a8z2 - 6a8z4 + 5a8z6 - a8z8 + a9z-1 - 8a9z + 15a9z3 - 13a9z5 + 6a9z7 - a9z9 - a10 + 9a10z2 - 15a10z4 + 10a10z6 - 2a10z8 + a11z-1 - 8a11z + 19a11z3 - 15a11z5 + 6a11z7 - a11z9 + 4a12z2 - 6a12z4 + 4a12z6 - a12z8 + a13z - 4a13z3 + 3a13z5 - a13z7 - 3a14z2 + 2a14z4 - a14z6 - a15z + a15z3 - a15z5 + a16z2 - a16z4 + 2a17z - a17z3

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 = -6                     1 j = -8                   1 1 j = -10                 1     j = -12               1 1     j = -14             2 1       j = -16           1 1         j = -18         2 2           j = -20       1 1             j = -22     1 2               j = -24     1                 j = -26 1 1                   j = -28 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, 114]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 114]]
Out[4]=	PD[X[12, 1, 13, 2], X[20, 9, 11, 10], X[14, 3, 15, 4], X[16, 5, 17, 6], > X[18, 7, 19, 8], X[4, 15, 5, 16], X[6, 17, 7, 18], X[8, 19, 9, 20], > X[2, 11, 3, 12], X[10, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 3, -6, 4, -7, 5, -8, 2, -10}, > {9, -1, 10, -3, 6, -4, 7, -5, 8, -2}]
In[6]:=	Jones[L][q]
Out[6]=	-(27/2) -(25/2) 2 3 3 3 3 2 2 -q + q - ----- + ----- - ----- + ----- - ----- + ----- - ----- + 23/2 21/2 19/2 17/2 15/2 13/2 11/2 q q q q q q q -(9/2) -(7/2) > q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-40 -38 -36 -34 -28 -26 -24 -22 -20 -16 -12 q + q + q + q - q + q + q + q + q + q + q
In[8]:=	HOMFLYPT[Link[10, Alternating, 114]][a, z]
Out[8]=	9 11 a a 7 9 11 7 3 9 3 11 3 -(--) + --- - 4 a z - 7 a z + 6 a z - 10 a z - 11 a z + 5 a z - z z 7 5 9 5 11 5 7 7 9 7 > 6 a z - 6 a z + a z - a z - a z
In[9]:=	Kauffman[Link[10, Alternating, 114]][a, z]
Out[9]=	9 11 10 a a 7 9 11 13 15 17 8 2 -a + -- + --- + 4 a z - 8 a z - 8 a z + a z - a z + 2 a z + a z + z z 10 2 12 2 14 2 16 2 7 3 9 3 11 3 > 9 a z + 4 a z - 3 a z + a z - 10 a z + 15 a z + 19 a z - 13 3 15 3 17 3 8 4 10 4 12 4 14 4 > 4 a z + a z - a z - 6 a z - 15 a z - 6 a z + 2 a z - 16 4 7 5 9 5 11 5 13 5 15 5 8 6 > a z + 6 a z - 13 a z - 15 a z + 3 a z - a z + 5 a z + 10 6 12 6 14 6 7 7 9 7 11 7 13 7 > 10 a z + 4 a z - a z - a z + 6 a z + 6 a z - a z - 8 8 10 8 12 8 9 9 11 9 > a z - 2 a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-8 -6 1 1 1 1 1 2 1 q + q + ------- + ------- + ------ + ------ + ------ + ------ + ------ + 28 10 26 10 26 9 24 8 22 8 22 7 20 7 q t q t q t q t q t q t q t 1 2 2 1 1 2 1 1 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 20 6 18 6 18 5 16 5 16 4 14 4 14 3 12 3 q t q t q t q t q t q t q t q t 1 1 1 > ------ + ------ + ---- 12 2 10 2 8 q t q t q t

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