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DrawMorseLink

PD Presentation:

X12,1,13,2 X8493 X14,5,15,6 X20,7,21,8 X16,13,17,14 X18,10,19,9 X10,11,1,12 X6,15,7,16 X4,21,5,22 X22,17,11,18 X2,20,3,19

Gauss Code:

{{1, -11, 2, -9, 3, -8, 4, -2, 6, -7}, {7, -1, 5, -3, 8, -5, 10, -6, 11, -4, 9, -10}}

Jones Polynomial:

- q-19/2 + 5q-17/2 - 12q-15/2 + 19q-13/2 - 25q-11/2 + 28q-9/2 - 28q-7/2 + 23q-5/2 - 17q-3/2 + 9q-1/2 - 4q1/2 + q3/2

A2 (sl(3)) Invariant:

q-28 - 3q-26 + 3q-24 - 3q-20 + 6q-18 - 4q-16 + 4q-14 - q-10 + 6q-8 - 4q-6 + 5q-4 - 2 + 2q2 - q4

HOMFLY-PT Polynomial:

az + 2az3 + az5 - a3z-1 - 5a3z - 5a3z3 - 3a3z5 - a3z7 + a5z-1 + 2a5z - 2a5z5 - a5z7 + a7z3 + a7z5

Kauffman Polynomial:

- z2 + 2z4 - z6 + 2az - 7az3 + 9az5 - 4az7 + a2z2 - 7a2z4 + 13a2z6 - 7a2z8 - a3z-1 + 8a3z - 18a3z3 + 13a3z5 + 5a3z7 - 7a3z9 + a4 + 5a4z2 - 31a4z4 + 43a4z6 - 15a4z8 - 3a4z10 - a5z-1 + 7a5z - 19a5z3 + 8a5z5 + 24a5z7 - 18a5z9 + 6a6z2 - 36a6z4 + 58a6z6 - 24a6z8 - 3a6z10 + 2a7z - 13a7z3 + 20a7z5 + 3a7z7 - 11a7z9 + 3a8z2 - 11a8z4 + 24a8z6 - 16a8z8 + a9z - 5a9z3 + 15a9z5 - 12a9z7 + 3a10z4 - 5a10z6 - a11z5

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 4                       1 j = 2                     3   j = 0                   6 1   j = -2                 11 3     j = -4               13 7       j = -6             15 10         j = -8           13 13           j = -10         12 15             j = -12       8 14               j = -14     4 11                 j = -16   1 8                   j = -18   4                     j = -20 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[11, Alternating, 352]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 352]]
Out[4]=	PD[X[12, 1, 13, 2], X[8, 4, 9, 3], X[14, 5, 15, 6], X[20, 7, 21, 8], > X[16, 13, 17, 14], X[18, 10, 19, 9], X[10, 11, 1, 12], X[6, 15, 7, 16], > X[4, 21, 5, 22], X[22, 17, 11, 18], X[2, 20, 3, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 2, -9, 3, -8, 4, -2, 6, -7}, > {7, -1, 5, -3, 8, -5, 10, -6, 11, -4, 9, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 5 12 19 25 28 28 23 17 -q + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q 9 3/2 > ------- - 4 Sqrt[q] + q Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 3 3 3 6 4 4 -10 6 4 5 2 4 -2 + q - --- + --- - --- + --- - --- + --- - q + -- - -- + -- + 2 q - q 26 24 20 18 16 14 8 6 4 q q q q q q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 352]][a, z]
Out[8]=	3 5 a a 3 5 3 3 3 7 3 5 -(--) + -- + a z - 5 a z + 2 a z + 2 a z - 5 a z + a z + a z - z z 3 5 5 5 7 5 3 7 5 7 > 3 a z - 2 a z + a z - a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 352]][a, z]
Out[9]=	3 5 4 a a 3 5 7 9 2 2 2 4 2 a - -- - -- + 2 a z + 8 a z + 7 a z + 2 a z + a z - z + a z + 5 a z + z z 6 2 8 2 3 3 3 5 3 7 3 9 3 > 6 a z + 3 a z - 7 a z - 18 a z - 19 a z - 13 a z - 5 a z + 4 2 4 4 4 6 4 8 4 10 4 5 > 2 z - 7 a z - 31 a z - 36 a z - 11 a z + 3 a z + 9 a z + 3 5 5 5 7 5 9 5 11 5 6 2 6 > 13 a z + 8 a z + 20 a z + 15 a z - a z - z + 13 a z + 4 6 6 6 8 6 10 6 7 3 7 5 7 > 43 a z + 58 a z + 24 a z - 5 a z - 4 a z + 5 a z + 24 a z + 7 7 9 7 2 8 4 8 6 8 8 8 3 9 > 3 a z - 12 a z - 7 a z - 15 a z - 24 a z - 16 a z - 7 a z - 5 9 7 9 4 10 6 10 > 18 a z - 11 a z - 3 a z - 3 a z
In[10]:=	Kh[L][q, t]
Out[10]=	7 11 1 4 1 8 4 11 8 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 20 8 18 7 16 7 16 6 14 6 14 5 12 5 q q q t q t q t q t q t q t q t 14 12 15 13 13 15 10 13 > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + 6 t + 12 4 10 4 10 3 8 3 8 2 6 2 6 4 q t q t q t q t q t q t q t q t 3 t 2 2 2 4 3 > --- + t + 3 q t + q t 2 q

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