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DrawMorseLink

PD Presentation:

X12,1,13,2 X2,13,3,14 X14,3,15,4 X16,5,17,6 X22,7,11,8 X9,18,10,19 X17,20,18,21 X19,10,20,1 X8,11,9,12 X4,15,5,16 X6,21,7,22

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

- q-44 + q-42 + q-40 + 2q-38 + 2q-36 - 2q-30 - q-28 + q-24 + 2q-22 + 2q-20 + q-18 + q-16

HOMFLY-PT Polynomial:

- a9z-1 - 17a9z - 36a9z3 - 28a9z5 - 9a9z7 - a9z9 + a11z-1 + 14a11z + 19a11z3 + 8a11z5 + a11z7 - 2a13z - a13z3

Kauffman Polynomial:

a9z-1 - 17a9z + 36a9z3 - 28a9z5 + 9a9z7 - a9z9 - a10 + 14a10z2 - 19a10z4 + 8a10z6 - a10z8 + a11z-1 - 15a11z + 33a11z3 - 27a11z5 + 9a11z7 - a11z9 + 13a12z2 - 19a12z4 + 8a12z6 - a12z8 - 2a13z - a14z4 - 2a15z + 2a15z3 - 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 = -8                     1 j = -10                     1 j = -12                 1     j = -14             1         j = -16           1 1 1       j = -18         1 1           j = -20       1 1 1           j = -22     1 1               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[11, NonAlternating, 254]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 254]]
Out[4]=	PD[X[12, 1, 13, 2], X[2, 13, 3, 14], X[14, 3, 15, 4], X[16, 5, 17, 6], > X[22, 7, 11, 8], X[9, 18, 10, 19], X[17, 20, 18, 21], X[19, 10, 20, 1], > X[8, 11, 9, 12], X[4, 15, 5, 16], X[6, 21, 7, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -10, 4, -11, 5, -9, -6, 8}, > {9, -1, 2, -3, 10, -4, -7, 6, -8, 7, 11, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(27/2) -(25/2) 2 2 -(19/2) -(17/2) -(13/2) -(9/2) -q + q - ----- + ----- - q + q - q - q 23/2 21/2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-44 -42 -40 2 2 2 -28 -24 2 2 -18 -16 -q + q + q + --- + --- - --- - q + q + --- + --- + q + q 38 36 30 22 20 q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 254]][a, z]
Out[8]=	9 11 a a 9 11 13 9 3 11 3 13 3 -(--) + --- - 17 a z + 14 a z - 2 a z - 36 a z + 19 a z - a z - z z 9 5 11 5 9 7 11 7 9 9 > 28 a z + 8 a z - 9 a z + a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 254]][a, z]
Out[9]=	9 11 10 a a 9 11 13 15 17 -a + -- + --- - 17 a z - 15 a z - 2 a z - 2 a z + 2 a z + z z 10 2 12 2 16 2 9 3 11 3 15 3 17 3 > 14 a z + 13 a z + a z + 36 a z + 33 a z + 2 a z - a z - 10 4 12 4 14 4 16 4 9 5 11 5 15 5 > 19 a z - 19 a z - a z - a z - 28 a z - 27 a z - a z + 10 6 12 6 9 7 11 7 10 8 12 8 9 9 11 9 > 8 a z + 8 a z + 9 a z + 9 a z - a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-10 -8 1 1 1 1 1 1 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 1 1 1 1 1 1 1 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 20 6 18 6 20 5 18 5 16 5 16 4 14 4 16 3 q t q t q t q t q t q t q t q t 1 > ------ 12 2 q t

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